Not sure if anything has changed since Oct 2021, but this says an iOS app cannot set focus mode.
However, MacOS can run a shell script containing “shortcuts run (whatever)”, and a shortcut can set focus mode. I have an always-on MacBook running Node Red, which receives an MQTT message from HA to run the shortcut. Clunky, but it works.
EDIT: This shows how an iOS app can run a shortcut, so I guess the companion app could be configured to run a helper shortcut to set the focus mode.