My MIL is like this. Probably not quite as bad, and I could write for hours with my own examples, but won't, just needless to say, she is very very similar. It also drove me potty.
I was similar, wanting to treat her and my parents the same. My advice is, don't. Your relationship with her has changed because of this behaviour. So adapt your behaviour. This is what I did - decided it was my DH's place to keep in touch with her, arrange visits etc, unless she contacted me directly. His place to send his parents photos. I sent to mine. The distance grew. She also does "he gets that from me/my son" etc and I let a lot of it go. I still argue back sometimes with "well I did..." as I can't help myself, but the bottom line is, you know he's as much yours as DH's. So does she. Try and sit comfortably with that knowledge, just smile, don't engage it.
My DC's are a bit older now - both primary school age. And what comes as they get older, is they will bump themselves, be feeling clingy, feeling sad, and in her presence they will come to me first and foremost. MIL will try to step in, but actually, THEY won't let it happen as they want me. That will come to you too, and is more powerful and settling for you than any verbal fighting back you can do (no matter how passive you try to make it). Those actions speak louder than anything and my MIL has been forced to accept she is grandma, not mum.
I know exactly how frustrated you will feel, but try and stay calm, if he won't settle for her, firmly say, "I'll take him, sometimes he just needs mummy" with a smile and gentle reminder of her place. She's overstepping, so don't feel uncomfortable reminding her of your place.
As for staying at hers, well ultimately, if you're not comfortable that won't happen. Say no, you and DH aren't going to split up over it (unless there are other issues?). If you're not comfortable you're not comfortable. End of conversation. You know him best, but might find if you manage to create some distance that in a year you are grateful for a night off.
What I try to keep in mind is that my children are very loved by MIL and that is a lovely thing. Now they're older and I'm more self confident of my position compared to hers, it's lovely to know when they stay there they are doted on, loved and happy. It will get better, just don't enable her behaviour.