WTBH......just want to tell you how I handled the birthday thing (as well as other special days)......
I did not include their father (my ex husband).....(he was a difficult narcissist).
I had the parties, etc.....and, the kids were very happy on those occasions.
If their father did special things for them on their visits to him...I am sure that they enjoyed it, there, also.
They did not seem to be "racked
by the idea that both parents were not in the same room---Norman Rockwell style...lol....
I think that is a fantasy dream that resides in the mind of the parent.....
Oh, sure...there are some people who are able to co-parent, together, beautifully....and have respectful, co-operative interactions with each other....
Those are normal, Healthy people.....
Your ex does not seem to fall into that category....and, may never be so....
Sometimes, you just have to accept the reality of the situation...and make the best of it......
dandylion