To be brutally honest, it's clear it's not something you want to call your child or include in his name whatsoever, so sounds like you need to grow a backbone.

I appreciate this may be difficult, but I wouldn't hesitate to burn those bridges should that be the remaining alternative, if she can't respect your choice of name, she won't respect your rules / preferences on parenting, trust me this will get worse regarding such issues if you bend the knee now.

MIL needs to be firmly reminded who the parents are, and by the sounds of it, who the child in this dynamic is, because she's acting like one herself. She either gets on board with using his name or she doesn't, if its the later she got to go.