

You could go to lichess.org and create your own time controls. They let you stipulate an initial time and an increment.
For example you could have a game with a 30 second time for each player and then add 15 seconds for each move after that. Increment ranges go from 0 to 180 seconds.
15 seconds plus 5 seconds per move should be doable.
I believe the time would be banked for each move rather than a hard fixed limit for each move.
You could then create a custom game open to others, or make the game available by link to a friend.
I don’t know about repeating or being able to have very long playtimes, but I do know the MIDI (.mid) file format can play songs at extremely small sizes compared to mp3 as it effectively stores the sound like music on a sheet which is played by a synthesizer. Also the MOD file format allows samples to be recorded as a sort of blend of synth and recordings.