I've not been a participant in these (and doubt I ever will be), but I've seen a couple now. They definitely are a lot of fun to watch.
As to your question, the organizers of the conventions are well aware that people can go by several different names within the fandom. For example, when you register, you provide a name for your con badge. It can be just about anything (doesn't have to be your username on forums/FA or specifically a fursona name, either). It could just be a random short string of binary if you really wanted. Basically, they don't really care as long as the name isn't inappropriate.
Given how they work things like that for registration, I imagine signing up for the dance comp would go about the same. Show up, give them a name, and that's that. Doesn't have to match the usernames you use anywhere, a fursona name, or even your con badge name.