I can't say I'm surprised at what happened. All these people being summonsed to various meetings, SIL obviously thinks herself a cut above the rest, let alone the 2-faced MIL.
As everyone else has said and as you yourself recognise, it's probably best to keep completely out of any family childcare arrangements. How hurtful though that MIL didn't feel she could say to you that looking after DS 2 afternoons isn't what she wants to do and had to bitch about it instead. As you say, she's entitled to her opinions but it must stick in your throat when it's work you're talking about whilst at the same time she is apparently happy to look after SIL's kids for weeks on end, so they can go on holiday as a couple.
And what a nasty person that shows SIL to be ..... she may have provided somewhere for your MIL to live but by the sounds of it that's no way guaranteed for the future anyhow and in the meantime, MIL is emotionally blackmailed. You can see why she'd want to keep them sweet but why that also has to involve slagging you off I don't know. Sounds like she's a weak person.
I personally wouldn't ever babysit SIL's kids again. They've already had huge favours from you, yet this has blown up because you've said a rare "no". Said in a "hurtful manner" ..... what the heck is that about ? Were you supposed to weep and wail at your inability to be properly servile to her and wear a cloth shirt ? She sounds like a spoilt madam who's used to getting what she wants, when she wants and doesn't consider other people whatsoever. From her reaction it's obvious she wasn't asking in the accepted sense, she was telling you that you were going to have her kids 3 weeks in a row.
Well maybe it's about time she woke up to the real world where most of us barely ever manage a weekend away, let alone 6 holidays a year on top without the kids as well. You have nothing to feel guilty about in this - you've done more than your fair share of enabling her to live it up ..... and if that means she won't sit your kids, well, it doesn't sound like you'd notice TBH.
You sound like a lovely caring person worrying about SIL's hurt but I think if you go bowing and scraping to her, acknowledging her "hurt" (over what exactly), you will only end up reinforcing the image she has of you as some inferior servant, who was in the wrong over this. If I were you I would be civil when our paths crossed, but you have nothing to make up and you shouldn't act as if you do.
SIL and MIL sound well suited to one another really. MIL's happy to effectively "sell-out" by dishing the "dirt" on you, feeding SIL's sense of superiority. SIL sounds completely up herself ..... she has no right to say she's "disappointed" in you because you were assertive ....... who the hell does she think she is, your bloody headmistress ?
I wonder how genuine talk of selling your mum's flat is ? Methinks last week's summons could have been an object in keeping her "on her toes", and making it clear which side her bread's buttered on. Nasty.
What you need to concentrate on now is finding childcare where DS will be happy. SIL's "hurt" doesn't matter. If you acknowledge that this superior attitude towards you will just continue. I'd pull right back from the lot of them.