Best friend baby mama is causing problems
The mother of my child and I are the best of friends, but this is making my current girlfriend very insecure. She now wants to dictate when and how I should talk to her as she says she doesn't trust me around her. I totally disagree with her request.
Is there a reason for her not to trust you particularly? It sounds like she does not have confidence in you and that's what you must work on. However, I am just imagining how it feels like for your partner to keep hearing you referring to your ex as your best friend. Can you imagine her always talking about her ex-boyfriend who is apparently her best friend? How would that make you feel? I can confirm to you that it is a different feeling when you have to be the one on the other side and you are asked to accept other people's unfair requests.
It is good to have a good relationship with the mother of your child but given the nature and history of your relationship, it will always be an issue for whoever is your girlfriend at that time. I would say keep the relationship civil, with good boundaries. Being best of friends will always work against you.
The mother of your child is your ex. Ex means she's in the past. Having a healthy and functional relationship with her for the sake of the child is to be applauded. However, getting close to her to the point where it's making your current lover insecure is unnecessary. Ask yourself this; if the tables were turned, would you approve? Do unto others as you would like it done to you.