It's time that the next time this woman invites the OP & DD over, she should start saying, "I can drop her off, but I can't stay." Of course, the woman will balk at the idea.

And cancel. That's to be expected. In counselling and behavioral modification we call that "Interrupting the Pattern." This woman has established a long pattern that has worked for
her that the OP conformed with. She's not going to want to give that up. She will cancel probably the first 2-3 times. But, after the third time, she may look at her DD playing alone & realize for her DD's sake, it might be better to have the friend come over, even if alone.
I think the OP has gone above & beyond what is required of her to keep a "friendship" going. This wasn't ever a
natural friendship. She was shamed into it by her mother.
If the OP is in her 50's then her mom is in her 70's. That generation of mothers wasn't into being your friend or your peer. They were the
mothers.

I was wondering that, too. Like maybe the reason that she's so tight with her money is not because she wants to be, but because her DH, the doctor, is controlling and a tightwade. He who holds the purse-strings (or in this case the wallet) controls the marriage.
For all we know, he may buy himself BMWs and is willing to furnish a beautiful home that reflects well on him, but is stingy to the point of lack when it comes to his wife & kids, so the DW
has to scrimp & save and make it appear to others that she
wants to.
Maybe her reason for avoiding going to the OPs house or letting her DDs go to movies with the OP is because she would then have to reciprocate those kinds of outings. Her DH may not give her enough money to go to movies.
I work with abused women. These things happen - even with doctors.