I think you need ot sit down with your DH and tell him a few home truths. As part of that, you have to let him know that the message attached to these home truths is - 'this is MY family too, especially now DD is here... and I am going to start putting my foot down and causing trouble unless you realise that your first loyalty is here, with us.'
You don't need to tell him you hate MIL to let him know just how unhappy you are with what's going on. You've seen MIL systematically turn the entire family, including him, against your SIL by weeping and wailing and bitching. You don't admire him for that. Tell him to sit down and think about how her manipulation has partially ruined those family relationships for his DD already - her future relationships with her aunt and cousins? Tell him to think about what exactly, if anything, SIL has done. And to think about the possible reasons for her pulling away from MIL in the first place which set all this off - when she pissed off MIL and MIL turned on her.
He can't think of anything? Well, you can help him on that. Ever since your DD wss born, you've had MIL as a constant, NEGATIVE presence in front of DD. Face Time twice a day, and never does she miss an opportunity to slag off DD's aunt and uncle in her presence. Any gifts seem to come with strings attached, and you never hear the end of them either. Now she is angling for long visits - and you're sad to say that your first thought is utter horror at the miserable experience that will be. Yes, DH, your own mum is already, within 9 months of your child's birth, making his wife feel exactly as you now imagine she made SIL feel - utterly fed up to the back teeth of the carping, the complaining, the joy-sucking. Where does he think that's going to end?
Tell him you are telling him this now so that something can be done, because you do not want things to end up just as bad between you and MIL as they are with her and SIL and you do not want to end up with DD not seeing as much of her grandmother because she is impossible to endure. You are cutting down the face time, as you are really concerned at DD being exposed to the bitchy, dysfunctional attitude MIL has. You are going to be asking him to step in more with contact fo the future, which hopefully will include him visiting with DD alone as she gets older, unless things improve. More than anything, you are saying NOW that there will be NO long visits - 3 days max, because you are now convinced that any longer, and you'll snap and then there'll be no going back. And - finally - now that your DD is here, there's going to be a new Christmas rule. No visitors staying over Christmas eve/day. There is NO WAY they are coming to stay over Christmas and especially for DD's first one. You can discuss how it's going to be long term and maybe things can change later - maybe one year at home, one to them, one to your parents, or maybe them coming a few days later -but you REFUSE to lose your first Christmas and that personal family time.
Set out your stall now.