Hi D.,
I've actually been on both ends of this spectrum. Wow, it's not easy for any age child when their parents begin to date other people. There is never a good age, but at two she will become extremely attached to anyone who she sees on a regular basis. My hubby and I have been married for 15 years. My oldest daughter did not meet him until I knew that he was serious and would be a great father to her. She asked him to adopt her. Have you asked your soon to be ex if he's serious about this woman, if so, then there's no problem for them to meet. I don't care if someone sees their child 10 or 90% of the time, they are that child's parent and if they are a loving parent and not abusive to the child then they have a right to see that child. Maybe your ex is trying to see how the kids and girlfriend "fit together". Which that is important to you as well, you don't want your child miserable. The more you accept the situation the more she will look at it in a positive light. Our children imulate their parents and they can tell when you're upset even if you're not "showing" it.
Now as for the girlfriend not wanting you and he spending alone time with the child, she doesn't have a say in that until she's married to him. BUT, once that marriage happens, she has a right to be at functions for your daughter as well. So it's important that you two get along or at least tolerate each other. This woman could be a great support for your daughter! We just don't know. I've been a step mom also...it's a tough spot to be in, but it can work if the adults decide to make it work! I'm not so sure that I would've been keen on the idea of alone time of my husband with his exwife. Look at it from both sides. Any way you look at this situation, the adults HAVE to compromise and agree to disagree sometimes and DON'T put the kids in the middle. I'm watching my step sister's little 5 year old daughter struggle through her parent's divorce. She is hurting badly and comes to her g'pa's house and doesn't want to leave. Not because she's abused by either parent, but because g'pa's house NEVER changes; it's her sanctuary. Divorce sucks and there will always be adjustments to be made. Good luck, I'm afraid there are no easy answers!