Your girlfriend's problem is that she's insecure about herself in relation to your proximity to other women of a similar age and who she sees as attractive. She's threatened by her ideas of them not by their actual existence. That's one of the reason's why she says she has no problem until you are in their physical presence. That and she's knows that her behavior is wrong and that she shouldn't be projecting those insecurities onto you and your innocent friends. There's nothing you can do about this, except ask her why she behaves like this and what her problem is. You're not responsible for your girlfriend's mental state. The only thing you're responsible for is sticking to the mutually agreed reasonable and realistic terms of your relationship and communicating how you feel with her about this behavior.

You shouldn't and really don't have to prove yourself to be believed. Unless there's some prior behavior you're not talking about here, any person demanding that you prove yourself is playing a losing battle. It's normally them who are behaving in the way they're accusing you of, or they have some left over resentment from a previous relationship that's rearing its head in your relationship.

What you're describing, not being allowed to have any female friends, that's called coercive control and it's not right and it has no place in a mutually supportive relationship. You do realize that you belong to you and that is who chooses who you spend time with. You don't have freedom if you have to be allowed to do something by your supposed equal. Your girlfriend will likely throw a fit, again, if you do contact this new friend. She's not OK with your choices and she's actively placating you by her current behavior. It's not talking bad to state to someone, like you have here, what actually happened. FWIW, I know you care for your girlfriend, likely love the version of her that exists in your head but ask yourself is this what you want the rest of your romantic life to be like? If it's not, think on what it is you want from and for your relationship to be about and if it's likely to happen with your girlfriend. If it's not, you know what to do.