… to want MIL to call our child by the name we gave her?

[fieryelephant]

Husband’s family / cultural circle (1970s- immigrants) have an ‘Indian-language’ name and an English name. i.e. Dennis Ganesha Smith. Most (including husband) now just use their English names. We invited MIL to chose for our daughter an Indian-language middle name. She chose a nice one. But, since birth, she has only ever called her by that name. I’m expected to refer to her by it. When others use her first / English name MIL can even say ‘who?’. It bugs me. Some days I feel like it shouldn’t, some days like it should. Obviously, it's not a practical problem, since everyone else uses the first name, but that doesn’t stop me getting unduly tense about this in our daily family life.
[The first / English name was, by the way, chosen specifically to be one that MIL would like, for family reasons, and she does.]

So accept that she would like to use a special name for your DD because she cares deeply about her but you don't have to put up with her asking "Who? " when others use her first name. That is rude and unpleasant and smacks of trying to permanently change the child's name.

I suggest you just say "You know perfectly well who." Every time she does that.