Yes, your MIL is being rude by not using your child's proper name when, I presume, she writes to him or her, but is this really worth having a family quarrel about?
If you follow some of the suggestions others have made here, you WILL have an almighty row with her. Her rudeness does not IMO entitle you to be just as rude.
I would try to ignore her rudeness after pointing out once again that letters or cards to the child are wrongly addressed.
You and your husband have decided that your child should have both your surnames, as you use the one you were born with. That is your business and a frequent choice these days, so your MIL is the unreasonable one here.
That said, don't let this blow up into a full-scale family row, it isn't worth it.