I think Max is a great, strong, stand alone name.
This. I have a grandpa Max and a 17-yr-old cousin Max, and it works just fine on both of them.
A 1-syllable LN might make me reconsider, but it's not a dealbreaker by any means if Max is your favorite. (Actually, my cousin Max has a 1-syllable last name, and I'd never noticed that until just now!)
And no, don't worry about celebrities naming their daughters Maxwell. That's silly :)