I think (but correct me if I'm assuming the wrong things) that your issue is not really just the cake, but that this is part of a situation of competing over who gets to do what for your daughter.
I have always been in a similar situation, although it is a bit easier as one set of grandparents is a long distance away. There is a constant dispute between my aunt and my mum as to who gets to do the big xmas event and they argue over my children, which is obviously madness as my kids have 7 great aunts so there is no reason why one in particular should expect to have them for xmas day.
My way of handling it is as follows (but wont work this year for you as you have already arranged your party), there is a 'family' event for my children of easter egg hunts, halloween parties, christmases, birthdays etc. My extended family argues between themselves over who is to host this as many of them want to. I don't host these ever.
Then there is what I consider to be the 'real' party, which at christmas means only me, ds, dd and dh on the day, and for every other event (birthday, halloween etc) means my children and their friends and I do not invite extended family as they want to do their own parties for my kids.
My children will refer to the stuff that goes on with grandma/aunts etc as their 'second christmas', third christmas and even one year their 'fifth christmas'. I think my kids do not mind at all having a fifth consecutive day of opening stockings and pulling crackers, or having 3 parties with 3 cakes, but I know they think of the 'real' party as being the one I organise, because they know I'm their Mum, even when my MIL has done a party first before my one on the actual birthday.
You need to find a solution you are happy with to mediate with your MIL and everyone else for the next 18 years, because I think things like this will come up over and over again. My method might not work for you, but I think it helps to find a way of distinguishing between your private family life and the extended family life.
So YANBU to feel the way you do, but you have to find a way of handling things that will suit you and make you happy.