I sympathise! My motto is that you can't pick your parents... and you can't always pick your metamours!
It's a tough balance, because often the hinge wants their two V arms to get along well. It makes things peaceful and less tricky. But - it's not essential to poly and hopefully he will be able to see that.
Women have a naturally competitive streak that is different to the traditional competitiveness of man. I can't remember what TV show I saw this on, but someone said that male lobsters in a boiling pot form a ladder to save each other - female lobsters drag each other down. I've seen it so many times and I think that there is some truth in it!
The best thing you can do here is remove yourself from her before you start to feel competitive. I can see that you are already starting to feel that you are.... better (bad choice of word, but it's the only thing I can think of)... than her in some ways: i.e. being more social, veering away from edgeplay, having more affection with Joe. It's natural that you would point this things out, because she is causing a hostile environment. When things are hostile, we automatically start to fight for survival of the fittest.
From what you've written, I think you've done a great job so far of trying to allow things to smooth over, without bending Joe's ear about her. If it were me, what I would do now is have an open talk with Joe - I'd tell him that you genuinely do not want to cause problems for him, or put him in the middle, that you would have liked to be close to her, but that you think your personalities are a better fit for a more casual friendship, than a close one. See what he says to that.
If this helps at all, my metamour has social awkwardness. Luckily, he is too uncomplicated to be malicious... ~laughs~... but he has a very, very low awareness of personal boundaries and appropriate behaviour. I live with him, along with my girlfriend who is the V, and I navigate him in the best way that I can, because I would prefer to live with them than not. Luckily for you, you don't have to live with her or be around her, so a cooler connection could work very well.
me: open poly (31, female)
GF: (41, female) my long-distance, long-term partner
Earth: (35, female) newly dating
Hubby: (38, male) GF's husband
Garcon: (28, male) GF's boyfriend/submissive
“Peace comes from within. Do not seek it without." ~ Buddha