"I desperately want DD to have a grandma and me to have a mum , but she's just so difficult.
I think this the issue you need to focus on.
You want DD to have the experiences and relationships we associate with grandmothers. Someone to take her to the park, or spoil her with sweeties, someone who loves and cares about it, perhaps especially someone with experience and wisdom. And in many ways you also want an adult daughter's relationship with a mother, that support, common sense, experience and love that an older woman who has already been a mum (and survived the experience!) who can cheer you on when things are wonderful and support you when they are difficult.
You like the idea of this person (or people). That is totally understandable, I think to some extent we all do - we are either glad we have her or sad that we don't.
The problem is that your mum isn't this person or people. Instead she is someone who wasn't properly there for you as a child - perhaps because she was badly damaged by her own upbringing, who still has issues as an adult. She isn't going to to be that iconic mother figure to you, nor that iconic grandmother to your daughter. Instead, she is hard work, manipulative etc.
So yes, if you had a mother figure who was lovely and kind and caring and wasn't going to make a visit at christmas stressful (well, more than usual festive stress anyway!), maybe someone who was able to help you with the cooking or present wrapping or shopping - or if they were too infirm to do those things, then if she was someone who you would still enjoy spending time with because she makes you laugh, makes you feel loved and appreciated. If your mother was like that, then yes it would be nice to invite her to stay. Nice for you, nice for her, nice for your DH and nice for your DD.
But if you invite someone selfish and hard work, someone who is only going to cause arguments and stress... that wouldn't be nice for at least three of the above people, perhaps all four.
You seem to have an accurate idea of who your mother is (manipulative etc). I think it is important to treat her as the person she is, rather than the person you'd like her to be (a good mum and grandma). I'm not saying that you have to drop contact if you don't want to, but I think that if you had just recently made contact with a different older woman who you had been out of contact with for years, who you knew to be hard work and manipulative, then you wouldn't be inviting her around for Christmas - you only feel you ought to because you are ascribing her with the status of a mother the person who you'd like to have in your life, rather than the person she is.
May I tentatively also suggest that those things that you would value in a mother for yourself, or a grandmother for your DD, that you find other ways of fulfilling those needs if it is likely that your mum will always fall short. It might mean doing some of the things you think grandparents would do as parents, or seeking to bring older women into your social life, perhaps neighbours, or through a church (or other relgious group if you attend/could attend one), or craft group, or some other activity/interest like a book, singing, history or writing group. (Maybe even the WI ;) ) Perhaps you'll find a regular childminder or babysitter for your DD that fulfills this role. If finding grandmothery type figures seems to difficult, then you might find that friendships that you can make with other mothers (through mother and toddler groups and other baby classes) might nourish that side of you that wants to feel connected to other mums, tap their wisdom and humour etc (Mumsnet is great for this, but in person is good too).
Sorry if that seems an odd suggestion, but when I lived away from relatives - before I became a mum even - I found that attending a craft group with a group of (older than me) women to be quite nourishing, in a way I hadn't expected (I was there for the crafts) and I also found that I appreciated my DH's family too. So I think there are things you can get from such friendships and relationships, that you associate with the role of a mother.