I think, mostly that it's the new girlfriend being unreasonable.
I don't think there should be any problem with you and the ex remaining friends, and doing things together sometimes with your DC. If she is seriously saying this can never happen because of the new child your ex is about to have, I think she is being very unreasonable.
However, the frequency and amount of contact you describe could be an issue.
DP stayed friends with his ex, they split up when their DS was 1. I came on the scene three years later, and was made to feel very welcome by DP's ex. We all get together (DP, me, DSS, ex, her partner, their child together, ex's stepchild from her DP, ex's mother, ex's stepfather - it's very 'blended'!) about once every couple of months for a BBQ, a dinner, a trip somewhere etc. We always have a little Christmas party together, though our actual Christmases are separate, and we always have a big birthday party altogether for DSS.
I actually have a great relationship with the ex, I really love her and her DP's daughter, and weirdly I have developed what is to me quite a special and important relationship with ex's mother. I love being part of a big blended family like this - for my own sake, as well as DSS's!
However. All that said, we get together about once every couple of months. Obviously DP sees them more frequently than that because he always does pick-ups. He might stay for a quick drink, but not more than 30m. What you seem to be describing (if I have understood) is that about 3 times a week your ex is spending a significant chunk of his evening with you and your child, either with coffee and chatting or dinner.
As happy as I am that DP is friends with his ex, and am completely non-threatened by their friendship, I would not be happy if he started spending time with her three evenings a week. It would just be too much for me. Partly because his working hours mean that quality time for us to be together during the week is precious, but partly just because...just because. I guess I would feel rejected and somehow isolated.
So I wonder if your ex's girlfriend has made this massive ultimatum because she was pushed a bit too far by the amount and regularity of contact still between you? On the other hand, it seems she's never wanted to get involved with any of it at all, and it's the mere fact of ANY contact rather than the amount that could be the problem. In which case...I think she is being very unreasonable. Maybe you and ex could agree to stop seeing so much of each other during the week, but get her to agree to do one joint trip/dinner or something all together just to test the waters and see how it goes?
Are you still single, by the way? Shouldn't really make a difference, but it probably does to her.