Forget the children. You need to discipline these grandparents! They are undermining you monstrously!
Choices:
Do you want to give her pocket money for her swimming lessons or a trip to the park?
Do you want to hold baby or read to DD?
Leave DD with them for a weekend. Pack ALL her chocolate to take away and not to be brought back. Wow, won't they regret that!
Also, film her when she's being bratty. Anything to expose them more to her nightmarish behaviour, so they see what they are leaving you with, and how unfair that is.
My MIL adores treating DS, and has often seemed quite hurt when I tell her it's too much, or it doesn't fit (oops), or he doesn't need something. However, she's really tried, and is much better now, proving that it can be done, and done without losing her darling GS's love.
Mind you, she still spoils him by doing everything for him - feeding with fingers and holding him to sleep included (!), but I am very interested to see that he doesn't love me, or my much-stricter-than-MIL mother any less than he loves her (MIL) despite our being stricter on him. I feel a bit sad for her, really, because he gives her much more trouble, but she is the one who doesn't have any confidence in her ability to discipline him without losing his love, so I guess I have to accept that, as long as it doens't affect my ability to handle him.
As for DD, is there anything else in her life which triggers her bad moods? I've discovered that television creates terrible problems for DS (3), as though that period of not-hearing-anything-else upsets his ability to concentrate and behave overall. When I remove the television (physically remove it) and we have only limited CBeebies through my computer, his mood throughout the entire day seems to improve; he's easier to manage at mealtimes and bedtime, too! I can't stress enough that it's not just when he is watching, and when it is turned off, but the entire day, which seems crazy, but there it is.
P.S. Sorry for the epic post, and apologies if none of the advice works for you, or if it's stuff you've already tried.