clemette - I think posting on AIBU is where you went wrong here (I have made the same mistake before lol)!
I understand what you mean about family politics. My mum is like your MIL. She introduced DS to swords and guns when she knew I didn't want her too (kept them banned from her own son till he was 8), she fills them full of sweets even if I am there and telling her not to, I try to keep them well-behaved and calm at family meals and she winds them up, tickles them and last time we went out was under the table with them tickling everybody's knees - including my 84 year old grandma's who was really embarrassed ! Bearing in mind this was about 9pm and they were too tired to go back to sensible once they got this hyped up! DD can take or leave gran (or anyone apart from me in fact) but DS is REALLY close to her and so constantly asks for her.
When we were going through family court and XP was having supervised contact she bullied me into letting her supervise it. I thought it would be a bad idea but she insisted and I gave in because really she was the easiest option. I had said to XP that he was not to ask for children past 3pm because the contact was difficult DS was only 18 months and dd 3 months and if he only came at 3pm it would make it hard for him and them as they'd be tired and upset going to bed and that would be the lasting memory of contact. She knew I'd said this and knew why and had seen how the children were the time he came at 3.30pm. He started calling her to arrange contact. One time he called at 4.30pm and she said "Oh you could come and see them now, we're not doing anything" I was listening and started shouting "No, I don't want him to come now" she kept offering and shushing me and then when she came off the phone she said "Oh don't worry I was only offering because I knew he wouldn't come now" I was and . How dare she think it was OK to go against a rule I had set and to take it upon herself to behave in a manipulative way with XP. This time was supposed to be about XP getting used to following simple rules and the motivation was to get him to see the children not manipulate him out of seeing them! We didn't speak for a while after this but have sorted it now.
I would never, ever rely on her for regular childcare. She is completely unable to respect things I have said or consider the consequences her actions will have on me and my family. She is a nice fun granny but her access has to be regulated for all our sakes! lol
In your situation I can see how it is vital to be political, MIL is a control freak, give her an inch, she'll take a few hundred miles. You can't stop her having involvement and don't want to because you all love her and she is part of your family. Stopping her from doing the one day a week and having a fight would not be a good idea.
I agree with what whereyouleftit said about using the nap as a reason to put the DCs in nursery care. "You are clearly getting tired having to provide childcare, I feel bad having that on my conscience, we want you to be able to be a gran and don't want to put on you." Then stand firm and repeat! Then offer alternative i.e. come and visit or take DCs out on X time to do X thing.