The easiest thing to ask / possibly change, is to ask her to eat somewhat slower, which might also help reduce with the burping. She may also be able to burp more discretely.

I suggest starting with this because it's a very simple concept-- instead of trying to juggle a lot of unfamiliar rules and muscle memory to use utensils differently, it is just conducting her normal routines with enough mindfulness she's slowed down some. Also: eating somewhat slower can be more healthy too, as you actually have a chance to chew food properly, which really starts off the digestion process.

Anyway, I'd recommend OP gently discuss with his gf her speed of eating, leaving the other things out of it, and just tell his mother to deal-- that he doesn't want to hear anything about how she eats ever again, and enforce that boundary.

But really-- this is a cultural difference, and I think it could possibly be toxic to this relationship of OP approaches this just assuming his habits are superior. They are culturally associated more with the rich and powerful, but this doesn't mean they are morally superior to a more rough and ready approach. If OP approaches his girlfriend with disdain or implies she is inferior to him for a cultural difference like this, it may poison their relationship.

Perhaps he can try eating with his hands when visiting her family, if he is going to ask her not to when eating with his on fancy occasions. (Copying a host's etiquette is actually pretty damn good manners, anyway.) If that's the case though, the wedding isn't the same ask as "please follow these habits in my parents house, we follow other habits in your parents house" because it is a jointly hosted event that belongs equally to both of them. Just cater a lot of food that's supposed to be eaten by hand lol.