Amen, I think you're showing a typically male attitude towards someone who is obviously in a vulnerable emotional state. With situations such a PND, it is not best to go in like a bull in a china shop arguing about Fathers rights. That is clearly not the issue.
Amen and Nellie - go to the Postnatal depression alliance (google the name it'll come up.) Sounds like you could both do with some information on PND and how to treat a sufferer. Nellie, I think this would be a good one to show your DH. I did the same for mine when I suffered and it did actually make him realise what I was experiencing.
Nellie, I can see that you feel clearly undermined by your MIL, and I think if any of us were in the same situation then we would feel the same way.
I don't know many mothers who would happily give up their small babies for the day knowing that she would probably be unsettled for the best part of it. First time Fathers often take a little time to get into the swing of things, and that is not a sexist comment, it's just the way it is. Same goes for first time mothers (of which you are not one.)
It's good you're going to the Docs about PND, this will help.
Re the Christmas situation, we had the same problem 4 years ago when our son was born. So I put my foot down and said we would do one GP's one year and one the other, and whoever we didn't see on Christmas day, we would duplicate that day on Boxing day.
To get the ball rolling you could suggest this to your DH ,with his mother being the first to have the Christmas day with you, and you go to your Mums on boxing day. Then when next year comes around, you have the ultimate bargaining tool. 'Ah yes, but we went to your Mother's last year. It's my Mother's this year. Fair is fair.'