I love where your mind is at for the big picture of the game. I can’t say much about what engine you should use but really anything from Twine to you making your own engine in HTML could work for this. Might I suggest checking out a few similar games first to help decide what you want for your own game.
A short list that might be helpful so you can get an idea of how the engine may effect your end product:
- You find a Fox - HTML
- Gorge - HTML (I think)
- Grope That Fox - Twine
There’s a lot fewer games of this genre than I expected tbh. These are just the ones I remember off the top of my head at the moment.