Hi Chris,
Long lived tokens do not work for this case. You need a signed path to access the content, AFAIK you cant create a signed path which never expires. So I created a node red flow which requests a signed path every time I need to access the media. It works perfectly.
[{"id":"970707c9e290e4c5","type":"tab","label":"Flow 1","disabled":false,"info":"","env":[]},{"id":"8a9451a1489823ee","type":"switch","z":"970707c9e290e4c5","name":"Short Press","property":"payload.event.args[0]","propertyType":"msg","rules":[{"t":"eq","v":"0","vt":"str"},{"t":"eq","v":"1","vt":"str"}],"checkall":"true","repair":false,"outputs":2,"x":170,"y":100,"wires":[["f95cdc2547c3b160"],[]]},{"id":"f95cdc2547c3b160","type":"ha-api","z":"970707c9e290e4c5","name":"Get AuthSign","server":"76b07c6c.a5b7a4","debugenabled":false,"protocol":"websocket","method":"get","path":"","data":"{ \"type\": \"auth/sign_path\", \"path\": \"/media/local/private/WhiteNoise.mp4\", \"expires\": 300 }","dataType":"jsonata","location":"payload","locationType":"msg","responseType":"json","x":390,"y":100,"wires":[["d9f6be0e73519a55"]]},{"id":"d9f6be0e73519a55","type":"function","z":"970707c9e290e4c5","name":"Make Payload","func":"varPath = \"http://192.168.1.190:8123\" + msg.payload.path;\nvarGoogleHome = \"music\";\nmsg.payload.data = {\"media_content_id\" : varPath, \"media_content_type\" : varGoogleHome, }\nreturn msg;\n","outputs":1,"noerr":0,"initialize":"","finalize":"","libs":[],"x":560,"y":100,"wires":[["c407630b2c25e128"]]},{"id":"c407630b2c25e128","type":"api-call-service","z":"970707c9e290e4c5","name":"Play White Noise","server":"76b07c6c.a5b7a4","version":1,"debugenabled":false,"service_domain":"media_player","service":"play_media","entityId":"media_player.googlehome7491","data":"payload.data","dataType":"jsonata","mergecontext":"","output_location":"","output_location_type":"none","mustacheAltTags":false,"x":750,"y":100,"wires":[[]]},{"id":"76b07c6c.a5b7a4","type":"server","name":"Home Assistant","legacy":false,"addon":false,"rejectUnauthorizedCerts":true,"ha_boolean":"y|yes|true|on|home|open","connectionDelay":true,"cacheJson":true}]