I really like the idea and the mouse is super cute and definitely the star of the show! :)
As already stated: If you wanna make it an actual rhythm game, you could prohibit left/right movement. I do think the obstacles are actually spawning on the beat, right?
The difficulty is pretty brutal, mostly due to the very sensitive controls. With just some tuning to values like the movement speed on input, jump height and fall velocity, I think the game would be a really nice play!