[Maybe things are a little (even a lot) awkward between them, but Chekov can't see Kitty in a dancing context and not offer. Dancing with her is fun. They're friends. They can still do fun friend things.
He sort of wants to go up to her and bow and maybe greet her the way that one of the characters in the old-fashioned movies that Sulu watches sometimes would, but approaching like a normal human just to make sure she's willing to dance with him feels safer.]
Kitty! [Nervousness doesn't make him less pleased to see her, anyway.] You are Kitty and not only someone with her face?