My MIL was a bit like this.
Ds went to an excellent CM, then MIL moved close to us & decided, when dd1 came along, that she'd like to look after her a couple of days a week. Great, but we had to negotiate with CM to have dd1 on the other days as this left her with a difficult to fill vacancy on MIL's days. Also, there were a few issues with MIL's ways not being mine...
Dd2 came along & I tried to suggest MIL would find a baby AND by then 2yo dd1 too much for her - would she like to just do babysitting? She was quite cross about this, insisted she'd be absolutely fine, looking forward to it, wanted same bond with dd2 as with dd1 etc etc.
Then SIL came to see me. She said MIL had privately admitted that it was all going to be too much for her. Fine, I said, quietly much relieved, & trotted off to tell MIL she could stand down, thank you so much for all you've already done, we'd make other plans.
MIL went ballistic - she'd said No Such Thing to SIL. Turned out SIL was expecting her own pfb & playing the long game - if her mum was doing free childcare for anyone, she intended it to be herself!
MIL & SIL had a huge row, then made it up & decided that whatever 'misunderstanding' had occurred was all obviously all my fault
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Anyway, the thing is, it was all an almighty PITA at the time, but now they're all at school, all have a lovely relationship with their grandmother, & I can't believe any of us allowed ourselves to get so wound up about it.
I'd suggest letting grandma stay, & doing a quiet family treat on another day entirely. Let the ongoing childcare arrangement potter along or unravel naturally if it's going to. It's just not worth a hoo-ha over it around ds's birthday, or it'll all get dragged up for years to come.