This looks very interesting. I am using Home Assistant with their companion Android app to track my location in the background (when on a charger/connected via Android Auto it will update more frequently). Would it be possible to integrate with HA so that it's not necessary to run another GPS tracking app in the background?

Gluetun works great for other docker containers. For regular apps Vopono is another great solution, which makes it easy to for example run another instance of a browser in a separate namespace.