I know I'm not supposed to play favorites, but I'm in love with this question. You have a beautiful mind, sir/miss.
My instinct is that Carrot Top would win in any fight because he's so ripped, but I'm intrigued to think of what distinct traits Carrot Bottom would bring to the table. First off, does Carrot Bottom have fluffy, green hair?
I'd like to think that the reason a person would get the name 'Carrot Bottom' is because they put a carrot in their bottom, which might prepare them for some hearty anal sex in the future, but not necessarily make them tough in a fight. Or, the name Carrot Bottom could refer to the position one likes while in sexual congress.
Any way you slice the carrot, I think the Top beats the Bottom. Unless of course Carrot Bottom is an hilarious comedian.
Short Answer: I think the question itself is stronger than the answer. I don't think I could possibly give an answer funny enough to do it justice. Well played.