You would not have tolerated any of this from a friend, family are no different.
You likely come from an emotionally healthy family where this type of familial dysfunction is thankfully unknown. Therefore this type of dynamic is unknown to you but what you are suggesting will not work out. A letter no matter how nicely worded, will be used against you somehow.
I would suggest you read "Toxic Inlaws" written by Susan Forward to further understand the dynamics being played out here.
What is your man's opinion on his mother's behaviour; are you really expected by him to tolerate this?. Does he say things like, "well you know what she is like". His own apparent inertia when it comes to his mother (is he really afraid of her deep down, it seems so) is hurting him as much as this is hurting you now. Both of you are in a fear, obligation and guilt state when it comes to her.
There is already conflict here but it is not necessarily of your making.
You are already tolerating her bullying behaviour of you by allowing her to look after your child twice a week. You are also tolerating having to toot your horn outside their house when you arrive there. Now you want to write her a letter saying that she could visit you Christmas morning for breakfast, this is really appeasement on your part and that never works out well.
What you need to do (along with your fiancé) is totally reassess and raise higher your boundaries (these are woefully low) with his mother. A united front needs to be presented. She is not and no longer rewarded for bad behaviour, if she cannot behave decently with either of you as this child's parents then she should see none of you. All the visits need to cease as of now. You will also need to find alternative childcare for your child twice a week.
It will also not do any children you have any favours at all for them to keep seeing you as their mum so disrespected all the time; children need decent role models as grandparents and his mother is patently not that person. Infact it would not surprise me if she was to use your son against you as his mother as well.