Your DH and MIL both want this, so you're best off giving in. Does it really matter if she's there? You could also invite her to stay over the night before. As regards her possibly arriving at 6, wouldn't you be up with the children anyway, with all their excitement? Or, does your DH get up at that time to put the bird in? Either way, if you don't want her to stay over and you also won't already be up, get your DH to open the door.
I might be biased, because I don't like the own-little-family trend at Christmas. My mother always shared us generously with her parents - they came to stay for a month at Xmas, living 350 miles away - and on Boxing Day all our cousins, aunts, uncles, and other set of grandparents would be over for the day. To me, Christmas is a multi-generational day, otherwise it's just like any ordinary day. I have a family member who hurt their parents very much with the own-little-family trend, so that's another reason I don't like it.
I don't think I'd have the heart to deny a grandma the chance to see them open their stockings if that's what they really wanted. I consider these relationships to be very important. And what would you possibly be doing with her not there that you can't do with her there?
Obviously, it's very nice for your DH to see his mum and his children together on Christmas morning. From his perspective, you're probably standing in the way of making those memories.
It's not worth the bad feeling between you and your DH. Let her come. Treat yourself to something delicious for breakfast.