I love f#. It's been so fun learning it and finding out that certain things like parsers and data transformation can be done so much more cleanly in a functional fashion. It's actually really changed the way I write c# now. I never realized how often I was essentially rolling my own error prone versions of Where/Fold/Select/etc via loops etc. Anecdotally I feel like I'm using at least 50% less loops these days.
F# works with monogame which should help you get going a lot faster than OpenTK. You have to kind of break out of the functional style to get the ball rolling but that's one of the great things about F#. It's functional by default but if you need you can make objects, mutable data etc.
It sounds like your game is somewhat event based and not super reliant on constantly updating values every frame so it might work out well. I'd probably shy away from doing something like a platformer in F# since you'd have to break out of a functional style so often it might to begin to not be worth it.
Here's a template for a monogame fsharp project.
https://github.com/romanov/monogame-fsharp/blob/master/Game1/Game1.fs