Hi S&G,
Since your MIL acted this way when she made you let her babysit, now you have a built in reason not to let her sit any more. Make her BS backfire. If she asks to do it, just quote the exact things she said. "No, we wouldn't dream of leaving you there alone while we go out." "No, let's wait until you can go out with wife to a movie or something. I wouldn't want to keep you from going out wth her when you two get a chance (when hell freezes over)". Etc. Throw her lying words in her face with a lovely smile and make her regret all her duplicity. And get what you want in the meanwhile.
How did you find out she was doing this? Did someone tell you? Then say, "I know you would never tell us this to our faces, but I understand that you were so hurt that we went out and left you here. We certainly would not repeat that! I'm so sorry we did it!"
About her complaining to all her friends about you: "I guess I'm just not up to par compared to your friends' sons-in-law. Listen, the very last thing I will do is impose on you to help us again." If she protests that this is really what se wants, say "I know that's what you said last time but honestly it wasn't fun for you, was it? I know better than to take advantage again." If she goes on about it, say "Let's not talk about it any more. I already feel a complete cad about what happened before. And now that I know you are too polite to be honest with us, I'm just going to have to put my foot down and refuse outright. Now let's change the subject."
It does not matter if she sees exactly what you are doing. In fact, it might be better if she does. Make sure you look happy while talking to her.
What do you have to lose?
Plucky