Your MIL and my DM would likely hit it off- but for the fact they’d both want to meet only in their respective favourite cafe!
My DM oscillates from narcissist to martyr, so if her manipulations meet with stonewalling from us she enjoys a full-on ‘poor me’ fest with anyone who’ll listen (hoping it’ll get back to us). She basically used to win either way. I think her record for whinging about me was 12 phone calls to other family in one afternoon!
Extending the ‘toddler tantrum’ analogy, we’ve learnt to manage her expectations very early. Makes us sound ridiculous to the outside world, but keeps us sane.
Bit the bullet and invited her to Christmas lunch at ours. Last August. Thus preempting her manipulation plan. Detailed email invitation to her, very positive, kind, yet matter-of-fact. Covered all bases about the lunch, presents, staying over. Every time she tried to pick holes in it, we just innocently re-forwarded the original email. Broken record technique.
She was enraged (didn’t even try to hide it) that we’d made it impossible for her to bitch about us (and our horrible treatment of her) to everybody. If we got a flying monkey, we just sent them the email too and didn’t engage.
She’ll still figure out a way to mess with Christmas, but we’ve avoided the insanity of the normal run-up. Sadly, our days of just hoping for genuine or loving treatment and twisting ourselves in knots to make her happy are over.
As others have said you didn’t create the hole inside your MIL, you aren’t responsible for making up for her crap life and truly you haven’t got the power to help her in any meaningful way. Only she can do that, and she’s chosen not to. Just aim for fairly peaceable, infrequent interactions and prioritise your own mental health.