Welcome! 
The team that makes The Handy are working on a new version of their API that supports websockets - they call it the “Realtime API Update”. When that comes out, then it’ll be possible to do exactly what you’re saying.
But yes, it’ll have to be integrated directly into the game.
I suppose it is possible to have the game synchronized if the game was matching the funscript - but it can’t be the other way around. There is a little bit of customization you can do since you can jump around within a single funscript - so you might be able to have, say, a two-hour funscript that has 24 5-minute patterns. That say, it would be possible for the game developer to simultaneously play a particular 5-minute animation track in the game, and skip to the correct section of the funscript that was designed to match that animation. But it’s probably not worth the effort…