Slim is the other one I'm looking at. in a just world I would be able to type "slim vs fat free framework" into google and see at least some forum posts or something comparing and contrasting the two, but all I get instead is top-ten lists that mention both without really giving a good comparison.
I've never heard of F3 so I can't compare and contrast, but Slim is definitely my go-to for projects like the one you describe. Just enough boilerplate to handle the basics, but not a ton of framework-specific quirks and conventions to learn.