Honestly the best way to find out whether you can handle a piece is to just try it.
I usually ask my teacher for advice in terms of where to go next though, since he has an unbiased eye and can sort of figure out what my natural technique and personality gravitates towards, rather than me convincing myself I'd do a better job learning a certain piece.
Funnily enough I'm on my way to finish off the moonlight, and when I listed a bunch of sonatas I'd consider next, he said "Tempest would be a good choice, only has a few tricky passages and would suit your personality and style the best in my opinion".
It's weird, I feel it's a great compliment to have someone even say I actually have a persona at the piano.
Anyway that tangent was meant to lead me back to the point that if you can comfortably manage the technical aspects of Moonlight, I don't see Appassionata being out of your reach technically either per se, but it will be very difficult, there is no such thing as a piece of music that isn't difficult. Music is inherently difficult. As long as you put in dedication and passion, you will get something out of tackling the sonata.
I say do it I don't see why not, if you're at the level of learning the 3rd movement of Moonlight I assume you have a teacher and so it's not like you'd perpetuate any sort of flawed execution by lkearning such a difficult new piece.