How does it sound if I told you that I'm not going to allow my mil to call my daughter by a nickname?
Oh wow, really? You are kidding me. Why won't you allow your mil to call your daughter by a nickname?
I think nicknames are immature.
I am sure she has good intentions. It is not good manners to tell your in laws what to do.
It's rude to not respect your daughter and call her a pet name.
The nickname won't harm your daughter I am sure, unless it is something like Duckface. Then I would worry but usually the names are just cute. Try to just let it all blow over.