I think she (mil) should be allowed to call your dd whatever she wants, within reason.I really strongly feel it's not worth getting het up about. Like you sadi, maybe when she's older your dd will be able to deicide for herself.
I went through something similar with my mil, but it was about her name rather than ds's. She wanted to be caleed 'nanny' which I loathe, I really do. Sorry, I know ther'll be plenty of people here who use it. It's really just a matter of what's used in your family/personal preference. I was brought up with 2 grandma's. So I thought ds would have 2 'grandmas' I really hate nanny and cringe every time I hear or have to say it.
I tried for the 1st month to enforce my preference, did get wound up about it. Bbut came to the conclusion it's her realtionship with him. She considers herself to be nanny, she wants to use it. Why the hell should I call the shots? So I let it go, accepted 'nanny' let her win her little battle.
But still cringe when I have to say it, lol.
I agree with the dogsbollox. OK, you might not agree with her conclusions, but this whole thing between you & mil is MUCH bigger than shortening your dd's name.
Maybe you should spend some time putting the name to one side and trying to work out what it is, then resolving this, or it'll be countless battles but the war will never end.