Update: You were all correct! I spoke to our L and there is absolutely nothing about my pregnancy that is relevant to any part of the court action. The second it comes up she will object based on that.
I'm guessing you expect that if his Ex knew your anticipated delivery date she would try to heighten the conflict about then trying to spoil the joy and distract from the event.
What would make you say that?
Yes, I do think she will disregulate over this. Although she is currently pregnant herself, I know she has grilled the kids for years about my intentions of having any children and they've known that was something I wasn't interested in and relayed that to her. I've never told the kids that anything has changed and their mom doesn't handle change unless she approves it first. She already tells the kids that I shouldn't have any kids because I'm too old. (I'm two years older than her.)
Plus we've found that any information she has becomes a weapon she uses in an argument. It would likely go something like: "So I can't have my summer visitation (that she is not even entitled to in court paperwork) when I want to just because that's when Nope is due? The kids don't need to be there for that because they told me that they don't care and they do have other siblings that live with me, you know. You're being selfish!"
It would be better if that could simply be avoided.