I think given the age of your child, the rawness of everything, and the fact you are going to your mum's for xmas, it is unrealistic of your ex to expect to have time with the baby at yours or his mum's house for Christmas this year. It just happens to fall on a saturday this year which makes it an issue for him.
The fact it is his birthday isn't particularly relevant. It will always be his birthday on xmas day and he can't expect to have your ds every christmas because of that fact.
Plenty of separated parents alternate the day or share it. And plenty go away to stay with family on their year with the children, or on their year without them too.
It's not compulsory you have to do it in any particular way unless ordered by a court, and given the amount of distress it will cause you and your family with things so new and raw it seems reasonable for him to leave you in peace this year. His mother can clearly see that too. He of course wants it all his way; I'll come to that in a moment.
So no you don't have to let him see your ds on Christmas day just because it's his only day off and his birthday too.
Yes in the future you should be looking at sharing things for your ds's sake. He has a right to a relationship with his dad. But at this point I think you need to be a little bit selfish for your sanity and well being.
Normal contact arrangements are generally suspended during school holidays and for Xmas and Easter even when children are not of school age. So you don't have to agree to it just because he's been coming to your place the last few saturdays. The reality is you won't be at home that day and it is a special day and things are different as a result.
Have you discussed with your ex's mother about an alternative day she (as grandmother) can see her grandchild? If your ex has promised her then maybe he is upset on her behalf. Perhaps you should try and come to an arrangement with her so that she can have a relationship with her grandchild despite her son and your relationship breaking down.
However you must bear in mind how your ex will be feeling. Are you simply punishing him for the pain he has and is caused you? Do you think he should stay with you because of the baby and because it's what you want and need regardless of the fact he may be miserable and unhappy by doing so? That's unrealistic and you need to accept it is over.
You have to let go of your relationship and move on for the sake of your child. It is not easy and it will take you a long time (we're talking years) before you realise this was probably a good thing and you are better off without him.
OK so firstly is your ex absolutely guaranteed to have Christmas day off even? A lot of restuarants insist that staff work on one of their busiest days of the year regardless of it being their day off. This may actually be a non issue if that is the case.
You also need to think about the fact that 2021 (I think) is the next time Xmas day falls on a Saturday. So you'll get every xmas day from now till then if your ex stays in the same job under the same conditions.
SO bearing that in mind is there no possibility at all that your ex could take the baby for a walk in his pram for half an hour perhaps? YOu could arrange to meet him slightly away from your parents house to avoid confrontation there.
Even if you are breastfeeding your baby can be away from you for half an hour. If you feel he may not bring the baby back then you could ask his mother to come with him.
A compromise like that could mean the world to him and may pave the way to a smoother parenting relationship in the future. YOu need your ex to establish that bond with his child that is so very important and that can only be done with time spent together.
Think about how you would feel if it was you not being allowed to see your ds your birthday or on Christmas day. Because if he takes you to court and the court orders alternate Christmases it could well be that you only see your ds on Christmas day every other year. How devastating would that be to you...? After all you are setting the precedent here by having the whole day yourself this year.
A very tough decision and I can understand completely why you may feel he doesn't deserveto see his ds and you can't handle it. However this is about your ds's right to see his dad, that is more important than what either you or his dad want or feel.
Best wishes, I know you are going through hell right now. It does get easier in time. A lot of time though :(