It's good that your DH recognises his Dsis can be immature and creates drama, and if he does fall out with his family it won't be your doing.
I really can't see why your MIL
doesn't tell her to give it a rest, after all your DH is her child too and if her own DD can't be civil to her sons wife she should tell her to grow up.
My mil pulled me to the side to say I was being extremely rude, I explained that sil hadn't spoken to me in a while & I did ask what was wrong
I've no doubt if there is a, genuine or not, reason for your SIL to treat you as she does your MIL will know what it is.
She said she didn't want to get involved
TBF your MIL isn't helping the situation either.
Despite what your MIL says to you she is getting involved by telling you you were rude to SIL but not saying why or giving a valid reason, so she must know.
Chances are the SIL is just not a nice person and knows your MIL won't hear a bad word said against her.
I'd have a word with your DH about future family gatherings, he knows how you feel and it can't be easy being pushed out for no know reason, then make a decision on the outcome.
If he's happy to not go fine, if you decide he goes you don't, again fine.
If he really wants you to join him do so but discuss beforehand that if you're made to feel unwelcome you both leave together, no drama, just a we're going now, say your goodbyes and leave.
If its mentioned by MIL/SIL as to why you left you're under no obligations to say why, the reasons will be pretty obvious to them why and you've tried to talk to them to discuss the ill feeling they didn’t want to know, so that ship has sailed