Welcome, Tempestseeker!
My heart goes out to you! You sound like a wonderful mum, MIL, and GM, and the fact that you have such unlimited access to most of your GC underscores this idea. I'm so sorry your relationship w/ P is not as good. It's certainly not for lack of trying on your part.
"Everything has to be on her terms ..."
I'm not sure what you mean by "on her terms." Perhaps she's more strict about schedules, etc., that your DD (dear daughter), DS, and XDIL? Some parents are very rigid about these things. It's wonderful if your AC and XDIL are more flexible, but flexibility may not be part of P's personality.
A couple of sentences or phrases jumped out of me that might provide some clues to the problem:
"...he cheated on my eldest grandson’s mother with her..."
IDKY you felt the need to point this out. Is it possible you have a negative attitude towards her/their relationship b/c of this? (I wouldn't blame you). And that she senses it despite your best efforts? If that may be, could you find it in your heart to let this past behavior go? Or, at least, move it a little further to the back of your mind?
"Everything has to be on her terms..."
I'm not sure what you mean by "on her terms." Perhaps she's rigid about schedules, etc? Some parents are. Your AC and XDIL may be more flexible (like my DD and SIL) and that's wonderful. But flexibility may not be part of P's personality.
Or maybe she's just a game-player as you and others have suggested. Regardless, IMO, the best idea may be to make your invitations through DS, as others have said. In fact, it may be a good idea to give him a list of dates and times that you're available for, say, a given month, let him go over it w/ P, and then get back to you about what will work for them. Chances are, P will have an excuse for every one of them, but IMO, it's worth a try.
If that doesn't work out, then I agree w/ other posters, that you should just make a casual invite, now and then, and leave it at that. Perhaps invite those GC to outings that you're taking w/ some of your other GC, so that even if DS and P say "no," you'll still enjoy a day w/ grands.
As for getting P to stop lying on social media, etc., I'm afraid I also agree w/ the others that you can't. Please just ignore her comments and don't worry what others will think. Those who know you will know it's not true, and those who don't, well, they don't really matter in your life, do they? Besides, not everyone will even be sure which GPs she's talking about. So please let it go as best you can and just enjoy the marvelous relationships you do have. xx