or do I not want to seem to be seen as backing down all the time.
She certainly doesn't need to see you backsliding. For a mother who has seen her children only a couple of hours (living 500 yards away) in five months......I would think she should be grateful that she gets any time with them through Christmas. They are old enough to express how they feel and apparently they have not wanted to see her during these five months or they could have done so, right?
Anyway, I think you need to be there for your kids and to heck with what she wants just b/c it's Christmas. She is trying to screw you over during the holidays....is my personal take on it. Be fair, but don't cave in. She needs to see you standing strong.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!