I am not trying to be insulting, but I am going to give you my perception of you bluntly. I don't believe that you are socially aware/intelligent enough to see interpret her behaviors to understand her motivations and handle the situation without issues.
If she is equally socially oblivious/ignorant, that complicates matters so much more because neither of you have any clue what the other is doing or thinking, which creates a big grey area that gets so much more messy than just one person being oblivious.
On the worse side is if she is very socially aware and intelligent, doesn't respect your relationship at all, and is wanting to manipulate things in her favor. She may just be with someone just so she isn't alone and he is just keeping her occupied while she waits for you.
People giving statements about their morality should always be taken to have an "unless..." tacked onto their statements. "I would never go after someone in a relationship, unless..." You may be an "unless".
The smart move is to never be alone with her. It sucks losing a friend because they have feelings, but keeping someone in the friend zone without feelings getting hurt or intentions misunderstood is hard enough when you are socially intelligent and aware, basically remote robotic rocket surgery while blindfolded if you aren't.
It is great that your girlfriend and you have an open dialogue about what is going on, that helps her feel more trusting of you and helps build confidence in the relationship and your fidelity. Talk with her and ask her what you should do about your friend.
I know I wouldn't be comfortable with my socially inept and oblivious girlfriend spending time with a guy who has made a move on her before and invited her to romantic activities alone on my birthday, but at the same time I would struggle with asking her to stop being friends with him. I would want them to stop being friends because I am insecure and have been cheated on and feel threatened by their friendship for fairly valid reasons.