I understand your MiL being 'annoyed'. Even an accident can be annoying, especially when it happens to something expensive and new.
But IMO she blew this all out of proportion.
Did she not see the babybels in your DCs hands? If she was so concerned about potential damage she should have put a blanket on the floor and said 'all snacks 'picnic style' on the blanket, please'. Besides, your Bro/Sis iL have DC so they're surely used to 'oopses' that result in stains. It's not like MiL was entertaining royalty.
You apologized and offered to pay for cleaning/replacement. What more could you do? Not sure how the 'discussion' descended into an argument, but what prompted her to disinvite you? Surely she didn't just say "your DC stained my new couch, you are banned!!". There must either have been something else said by one or both of you, or you and MiL have 'baggage' between you.
I think YWNBU to leave and take the children. I'd have done the same thing. No one insults me, bans me, and then can expect me to do favours like leave my children behind. And after your DH's comment, hell would freeze over before I'd drive to pick him up. He can catch a train, a bus, or hitchhike home. And if MiL wants to communicate with me, unless she wants to apologize for flying off the handle at an accident, at this point she can put it in an email. I wouldn't drive down just so she could lecture me.
Afterthought: if you do drive down to pick him up I'd let him know in no uncertain terms that I will NOT be lectured by MiL. That you will either honk for him and he can come out to the car OR you'll meet him at the nearest carpark.