Your husband needs to toughen up, frankly. And you need to toughen up with him.
He needs to tell mom and dad, no this overnight will not happen. No matter who picks up your child. Why was this agreed to? What made you or him say yes to this? Did you or he say yes before your MIL drank while caring for a child and is that why you're going ahead with this? If so -- that's not an adequate reason. Is he scared of offending his mom? Are you? It's time to do some offending, then.
There seems to be a good likelihood that MIL would be alone with the kids, your own and your niece. And your FIL is in DENIAL -- and someone in denial is going to think it's fine to leave his wife alone with two young kids. So you cannot really trust FIL either because his judgment is clouded. And this isn't just about the driving arrangements. Do you really want your kids sleeping over where she might be flaked out so hard she can't hear them in the night?
Your husband needs to be the one to tell his parents that he is no longer comfortable with the sleepover. If you and he have social plans and the sleepover makes those possible -- sorry, time to put the kids ahead of either those plans or your MIL's tender feelings.
Your husband sounds as if he too may be in some denial if he's letting this happen. You need to wake him up. Take him with you to an Al-Anon meeting ASAP. They are everywhere, all the time, just like AA meetings are -- under the radar but everywhere. He needs a wake-up call that he has to refuse to tiptoe around his mother's illness, especially if he has young children and she thinks she's going to see them.
He also must, must tell SIL. This should have been done already. If he won't, you should, but truly it is his role as the brother and son in this situation. For her child's safety she needs to know. She too may be in some denial and she may get angry -- "Are you saying I don't know my own mom? Are you saying I dont' know what's safe for my child?! It's fine" and so on....but she should be told. And she needs to know too that FIL is not seeing reality here and therefore she --and you--cannot rely on "It'll be OK as long as grandad is there too."
Husband's family is enabling mom to drink if they don't give her any consequences. If she wants to see the grandkids, she gets treatment. No treatment, no grandkids. Your husband or his sister can locate an AA meeting that works for her and then DRIVE HER THERE every time and wait outside until it is done, over and over. And her husband needs to go to Al-Anon to see how his blinders will only make it easier for her to drink.
You wrote "I feel like this entire family isn't taking MIL's drinking seriously." So what do you plan to DO about that? Because she is going to keep wanting to see your kids. Do you really want to spend years worrying every time she sees her grandkids that THIS might be the time she falls asleep drunk and leaves the stove on with grandpa out of the house but the kids there? Or that this will be the one time she thinks it's fine to put your kids and niece into her car "just to drive a few blocks to the store" and she's actually drunk? Or that this will be the one time she isn't aware that the kids have gone out the back door on their own to who knows where?
It only takes one time - just one time for someone to get injured or killed. And even if these kids never get a scratch, what do they learn from growing up with alcoholic grandma? They'll learn to look the other way -- like your husband has.
I would sit husband down, when the kids are not around, and he is not wanting to go somewhere else or do something else. Make clear to him that this is a very serious talk and at the end of it some decisions have to be reached. Before you talk to him, have Al-Anon information (go online) to talk about what enabling is, and why it gives his mom a green light to drink if he and his dad are as soft as they are right now towards her. Do not attack her as weak or bad because she isn't -- she is ill. But her illness could cost a child greatly.
If the sleepover is this weekend you might have to be the bad guy -- with your husband, when you tell him that you AND he cannot allow this, and cannot rely on FIL either.