Ok, so it has taken some time to get around to this, but here are the results.
I set logger default to debug, and then used tail -f on the log file to see what is going on real time.
When I use the toggle button from the browser, I get the following output immediately, and also immediate light change:
2019-01-26 04:43:38 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Received {'service': 'turn_off', 'type': 'call_service', 'service_data': {'entity_id': 'light.master_bedroom_closet_light'}, 'domain': 'light', 'id': 32}
2019-01-26 04:43:38 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event call_service[L]: service=turn_off, service_data=entity_id=light.master_bedroom_closet_light, domain=light>
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.light.lutron_caseta_pro] Writing light OUTPUT value: 8 1 0 None
2019-01-26 04:43:38 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'result': None, 'type': 'result', 'id': 32, 'success': True}
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Read value for host 192.168.1.11: OUTPUT 8 1 0.000000
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.light.lutron_caseta_pro] Got light OUTPUT value: OUTPUT 8 1 0.000000
2019-01-26 04:43:38 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event state_changed[L]: old_state=<state light.master_bedroom_closet_light=on; supported_features=33, area_name=Master Bedroom, friendly_name=Master Bedroom Closet Light, brightness=255, integration_id=8 @ 2019-01-25T20:43:17.749694-08:00>, new_state=<state light.master_bedroom_closet_light=off; supported_features=33, friendly_name=Master Bedroom Closet Light, integration_id=8, area_name=Master Bedroom @ 2019-01-25T20:43:38.252674-08:00>, entity_id=light.master_bedroom_closet_light>
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:43:38 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'type': 'event', 'event': {'data': {'old_state': <state light.master_bedroom_closet_light=on; supported_features=33, area_name=Master Bedroom, friendly_name=Master Bedroom Closet Light, brightness=255, integration_id=8 @ 2019-01-25T20:43:17.749694-08:00>, 'new_state': <state light.master_bedroom_closet_light=off; supported_features=33, friendly_name=Master Bedroom Closet Light, integration_id=8, area_name=Master Bedroom @ 2019-01-25T20:43:38.252674-08:00>, 'entity_id': 'light.master_bedroom_closet_light'}, 'context': {'user_id': '98f89f7ea7d94be1a36ecd6d92de583a', 'id': 'ab3873379e2345f098c75e5db08ac351'}, 'origin': 'LOCAL', 'time_fired': datetime.datetime(2019, 1, 26, 4, 43, 38, 252954, tzinfo=<UTC>), 'event_type': 'state_changed'}, 'id': 2}
Alternatively, if I push a remote button, I get the following output immediately, but wait another 1-2 seconds before the light transitions. There is no additional output just before or after the light transitions:
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Read value for host 192.168.1.11: DEVICE 6 4 3.000000
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.sensor.lutron_caseta_pro] Got DEVICE value: DEVICE 6 4 3.000000
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.sensor.lutron_caseta_pro] Got Button Press, updating value
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event state_changed[L]: old_state=<state sensor.master_bedroom_bedside_remote=0; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:41:54.263824-08:00>, new_state=<state sensor.master_bedroom_bedside_remote=4; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.178074-08:00>, entity_id=sensor.master_bedroom_bedside_remote>
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'type': 'event', 'event': {'data': {'old_state': <state sensor.master_bedroom_bedside_remote=0; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:41:54.263824-08:00>, 'new_state': <state sensor.master_bedroom_bedside_remote=4; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.178074-08:00>, 'entity_id': 'sensor.master_bedroom_bedside_remote'}, 'context': {'user_id': None, 'id': 'af8f3010e81b4877983e6be1e75710f0'}, 'origin': 'LOCAL', 'time_fired': datetime.datetime(2019, 1, 26, 4, 42, 0, 178303, tzinfo=<UTC>), 'event_type': 'state_changed'}, 'id': 2}
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event automation_triggered[L]: entity_id=automation.bedside_remote_off_press, name=Bedside Remote Off Press>
2019-01-26 04:42:00 INFO (MainThread) [homeassistant.components.automation] Executing Bedside Remote Off Press
2019-01-26 04:42:00 INFO (MainThread) [homeassistant.helpers.script] Script Bedside Remote Off Press: Running script
2019-01-26 04:42:00 INFO (MainThread) [homeassistant.helpers.script] Script Bedside Remote Off Press: Executing step call service
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event call_service[L]: service=turn_off, service_data=entity_id=['light.master_bedroom_closet_light'], transition=1, domain=light>
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.light.lutron_caseta_pro] Writing light OUTPUT value: 8 1 0 1.00
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event state_changed[L]: old_state=<state automation.bedside_remote_off_press=on; friendly_name=Bedside Remote Off Press, id=bedside_remote_off_press, last_triggered=2019-01-25T20:41:51.821066-08:00 @ 2019-01-25T20:31:20.799238-08:00>, new_state=<state automation.bedside_remote_off_press=on; friendly_name=Bedside Remote Off Press, id=bedside_remote_off_press, last_triggered=2019-01-25T20:42:00.242538-08:00 @ 2019-01-25T20:31:20.799238-08:00>, entity_id=automation.bedside_remote_off_press>
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'type': 'event', 'event': {'data': {'old_state': <state automation.bedside_remote_off_press=on; friendly_name=Bedside Remote Off Press, id=bedside_remote_off_press, last_triggered=2019-01-25T20:41:51.821066-08:00 @ 2019-01-25T20:31:20.799238-08:00>, 'new_state': <state automation.bedside_remote_off_press=on; friendly_name=Bedside Remote Off Press, id=bedside_remote_off_press, last_triggered=2019-01-25T20:42:00.242538-08:00 @ 2019-01-25T20:31:20.799238-08:00>, 'entity_id': 'automation.bedside_remote_off_press'}, 'context': {'user_id': None, 'id': 'af8f3010e81b4877983e6be1e75710f0'}, 'origin': 'LOCAL', 'time_fired': datetime.datetime(2019, 1, 26, 4, 42, 0, 243773, tzinfo=<UTC>), 'event_type': 'state_changed'}, 'id': 2}
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Read value for host 192.168.1.11: OUTPUT 8 1 0.000000
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.light.lutron_caseta_pro] Got light OUTPUT value: OUTPUT 8 1 0.000000
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event state_changed[L]: old_state=<state light.master_bedroom_closet_light=on; supported_features=33, area_name=Master Bedroom, friendly_name=Master Bedroom Closet Light, brightness=255, integration_id=8 @ 2019-01-25T20:41:54.212864-08:00>, new_state=<state light.master_bedroom_closet_light=off; supported_features=33, friendly_name=Master Bedroom Closet Light, integration_id=8, area_name=Master Bedroom @ 2019-01-25T20:42:00.278128-08:00>, entity_id=light.master_bedroom_closet_light>
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'type': 'event', 'event': {'data': {'old_state': <state light.master_bedroom_closet_light=on; supported_features=33, area_name=Master Bedroom, friendly_name=Master Bedroom Closet Light, brightness=255, integration_id=8 @ 2019-01-25T20:41:54.212864-08:00>, 'new_state': <state light.master_bedroom_closet_light=off; supported_features=33, friendly_name=Master Bedroom Closet Light, integration_id=8, area_name=Master Bedroom @ 2019-01-25T20:42:00.278128-08:00>, 'entity_id': 'light.master_bedroom_closet_light'}, 'context': {'user_id': None, 'id': 'af8f3010e81b4877983e6be1e75710f0'}, 'origin': 'LOCAL', 'time_fired': datetime.datetime(2019, 1, 26, 4, 42, 0, 278362, tzinfo=<UTC>), 'event_type': 'state_changed'}, 'id': 2}
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Read value for host 192.168.1.11: DEVICE 6 4 4.000000
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.lutron_caseta_pro] Invoking callback for host 192.168.1.11
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.sensor.lutron_caseta_pro] Got DEVICE value: DEVICE 6 4 4.000000
2019-01-26 04:42:00 DEBUG (MainThread) [custom_components.sensor.lutron_caseta_pro] Got Button Release, updating value
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.core] Bus:Handling <Event state_changed[L]: old_state=<state sensor.master_bedroom_bedside_remote=4; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.178074-08:00>, new_state=<state sensor.master_bedroom_bedside_remote=0; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.405522-08:00>, entity_id=sensor.master_bedroom_bedside_remote>
2019-01-26 04:42:00 DEBUG (MainThread) [homeassistant.components.websocket_api.http.connection.1974483696] Sending {'type': 'event', 'event': {'data': {'old_state': <state sensor.master_bedroom_bedside_remote=4; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.178074-08:00>, 'new_state': <state sensor.master_bedroom_bedside_remote=0; friendly_name=Master Bedroom Bedside Remote, integration_id=6, area_name=Master Bedroom @ 2019-01-25T20:42:00.405522-08:00>, 'entity_id': 'sensor.master_bedroom_bedside_remote'}, 'context': {'user_id': None, 'id': 'ac47d6b212954190bffa99cdc550e1df'}, 'origin': 'LOCAL', 'time_fired': datetime.datetime(2019, 1, 26, 4, 42, 0, 405685, tzinfo=<UTC>), 'event_type': 'state_changed'}, 'id': 2}