Its like she loves causing an argument with DD which causes DD and me to have a heated discussion.
Ethel, the above statement needs clarification....You say that you and DD end up having a heated discussion. Does your counterpart somehow goad you and your DD to argue with each other? Has your DD ever acknowledged that her MIL is adversarial towards you?
Without a clear understanding about what's really going on, it's hard to give advice, but you sound like a kind person who isn't in it for a fight. If you ignore her rudeness and take the higher ground, she'll have to come around sooner or later. You both have a lot in common. You are now part of the same family, the two of you love your (shared) grandchildren dearly and you will continue to love them for the rest of your lives. The negative impact this is having on your DGC's birthday's seems unfair and carrying this feud years into the future would be daunting at best. From my limited perspective, I see an insecure grandmother who is afraid to lose her family's affection. I do hope that with a positive, loving intervention, things can improve for both of you. You both want the same things and I'm sure there's plenty of love to go around.