I'm not sure my advice is worth much at all - I'm not having success in my own sitch yet!
But... it seems she's given you some interesting feedback on how she perceives your sitch - she feels enormously pressured and this seems to be related to what she perceives as pursuit on your part.
So, although hurtful (and wrong from your POV), this is what she feels.
Given that, and given that your goal is to show your changes, i'd say let her take him for the day. It's a special day (birthday) and probably wasn't planned in advance.
It shows that you are the bigger person, and can be flexible (this will pay off for you in the long run, too, I'd imagine).
And as you are not yet working from court-imposed visitation orders, no need to act as if you are.
I know it seems unfair and wrong to act like a doormat - but.... which action is likely to get you closer to your goal of showing consistent change?
Also, it means you get to see her when you drop off and pick up that day. Two more opportunities to show the new you.
