I have DD who is Catherine - DH prefered this spelling and I didn't really mind which way it was spelled! We still shorten to Katie with no problems.
I have a friend who is Katharine and she said she has spent her life having to correct people and it drives her mad! (Katharine Hepburn has this spelling and was who her Dad decided to name her after!) Do you daughter a favour and spell it either Katherine or Catherine, then all you get asked is 'is that with a C or a K?'