Personally, I love Katherine (with the K)... like Katherine Hepburn. I am not as much of a fan of it with a C. But that is just my personal preference. Generally speaking, I much prefer "traditional" spellings. If you absolutely cannot stand, just really ABHOR it with a K, then maybe you should pick another name altogether.
I have generally found, over the years (we are coming up on 15th wedding anniversary pretty quickly here), that in matters that really have no consequence (and often in some that do, lol), as in nobody is going to die because of the decision one way or the other....I mean... it really won't tangibly affect your DAUGHTER whether her name is spelled with a C or a K... that when I defer to my husband, I am usually glad of it later on. He's a pretty good man, and one of the vast number of reasons I married him is because I respect him and his opinions.... sooo.....
It's kind of one of those things of "do you really want a battle on THIS hill"? If you cannot abide the name... absolutely CANNOT... then pick something else. Otherwise, let him choose the spelling.
That's my my opinion...
So that you know I am not just spouting off about something that I can't relate to, though, I will tell you that our first child has a name that I NEVER would have chosen. In fact, we thought he was a girl, then found out 3 days before the birth that nope, he's a HE. Hubby wanted the name "Aaron" I said "Erin" would have been okay for a girl, but NO WAY did I want a boy named "Aaron". Guess what my son's name is? And I LOVE it. :)
p.s.
For what it is worth: to me, when I see Catherine with a C, I think of a soft spoken, mild-mannered, quiet, demure lady. When I see Katherine with a K, I think of a self-assured, self-possessed woman who while modest, is more aggressive and outspoken about her abilities and opinions. JUST what I think....