I ran into an issue getting sound using mopidy over bluetooth on a raspberry pi, but this link resolved it (which it seems you have already tried): Running as a service — Mopidy 3.1.1 documentation
From your description, the mopidy as a service does play sound over bluetooth for ~4 seconds. Does the tunein stream work for longer when playing using a audio jack? Does the sound also cut out when playing a local music file? Is the bluetooth speaker still connected after the tunein stream drops?