OP. The timing of this lunch is no accident. Why is it so urgent it absolutely has to happen this weekend? It's not a lovely peacemaking occasion, although they will all keep saying that it is ( although you haven't declared war, you've simply said no to some childcare.) Its CF Phase 2
They have already used PILs, and Bil to BIL to pressurise you. trying to get your own DH to TELL you to do it. Is it that unreasonable to assume that both sets of ILs are looking to strong arm you or at the very least, put you in your place?
They will keep on at you and DH because they know you are a kind nice person and they want you to do what they want. Family loyalty, Oh but the Kids, Surely we are here to help each other, etc.. They do not care about ANY of your genuine reasons ( or as they call them excuses) Not one.
They want you to solve their problem come what may. Both sets do... Otherwise why isn't dear MIL offering to cover.? She's going to say she's doing her bit by having the lunch! Getting the family "together". She may well come up with some kind of "proposal". which will be hard to turn down because on the surface it's "meeting you halfway" (nevermind halfway to what?).
They will use the fact that the kids are there, to play on your heart strings, look over at them fondly and say "let's not argue in front of the kids" (even though you are not arguing.) "You see how well they get on?" "Let's set a good example (do what we want)" and then look meaningfully at you. If you disagree, you will be the Bad Guy. You already are even though you have already generously helped them, they all conveniently forget that. BIL and SIL consider themselves the victims here. Quite Trumpian really. You personally "upset SIL" remember? Nevermind the stress this pressure has caused you.
If you want them to stop taking advantage of you and disrespecting your time you have to say to yourself on repeat "Give them an inch and they will take a mile." These are grade A CFs and they don't care how much they have to lean on you to get their way. Additionally, they are really cross that you have dared to say no to them and will be keen to put you back in line.
Ironic really, if they put this much effort into finding their own childcare, they'd have it sorted by now.
Do not:
Send DH and kids on their own - they will try to convert him and strong arm him. Also. "We tried to talk reasonably to OP but she wouldn't even come to lunch. We are so upset at her attitude towards the family."
Do not: Go thinking that you can simply leave if it becomes unpleasant - It is much harder to walk out gracefully than you think and they will simply say,
"We tried to reason with OP but she stormed out in a temper of the lovely lunch (even if you didn't) we planned to bring the family together. She is impossible and the poor children were so upset that their Aunt doesn't want to spend time with them. Her DH had to be dragged out too and we could see he was upset "
Do not: agree to any suggestion of them having your kids on a Saturday or an evening in the holidays - or ask them to.. see giving an inch abpve . It will simply be used to fabricate an illusion they've helped you and now it's your turn. "Oh but we had your DC when... "
Your best bet is to keep them at arm's length, politely, until the holiday crisis is over.
You are busy but would love to meet up weekend of 7th September. (preferably in a park/picnic/outdoors activity scenario) as you really want the kids to have fun together. I think that this method will be the most peaceful solution as the lunch could get quite fraught and it will give time for things to cool down when the holiday pressure on both sides is removed as you won't have to be refusing them emergency help. At the moment they still think they can persuade you. I think this is the quieter way to cool things down.
I realise reading back over this that what I've said sounds very cynical and may look like harsh judgement of your in laws. You may be thinking It won't be as bad as that. You could well be right and perhaps I am being a bit cynical, but it's based on lived CF experience and also on your descriptions of what they have done so far. If there is a "CF Script" your SIL is following it.
But ultimately, you know the characters involved best to pick your best path. I wish you best of luck dealing with this. You are not an unkind person to stand up for yourself and your own DC.