Mother-in-law

I can't say we have the same situation, but you can compare it a little bit.
Here, the son goes to daycare every day, no (grand)parents who have a fixed day to babysit.
It's also difficult, grandparents live 2100 km away and my parents 65 km (my father has his own business).

Now it's also like this, that my mother asks very often 'when is the little one coming to stay over'.
When it comes down to it, it never works out or they have other plans. Or she ignores the issue completely and has to discuss it with my father.
My son was there last month for 3 nights, on the weekend. The first day (or earlier in the morning) that my mother was alone with my son (father had to work), she called 4 hours after he got up that we had to come pick him up.
We then also decided that he won't stay there anymore, my mother is no longer able to take care of him.
Even if she shouts from the rooftops that she wants to babysit, it just doesn't work anymore.

We can't put this on board, since my mother has a number of problems that she denies herself and that my father also denies). Saying that she is no longer capable of taking care of him, would break her.
I would just talk to your mother-in-law again. Ask her concretely which day she wants to babysit, and tell her that you completely understand if it's difficult for her. Maybe she can take a fixed day a month instead of a week? So that she feels that you want to include her but that she can determine at her own pace when it fits for her and when not. (E.g., every two weeks on Sunday)

Good luck anyway!