Sigh!
As always, tboy is trying to talk about something he has absolutely no concept of. Maurice, you're not helping much. Open relationship, be they more swinger type or more polyamorous type, are not the beginning of the end. But clear communication skills are a must if such a relationship is to survive. This is the root of the issue Bliss is dealing with.
Blissful, what you're experiencing is a breach of trust from your partner, and you don't seem to be communicating effectively. Your neighbour took your admission that you and Brash are open as license to insinuate herself into your relationship, without knowing the established boundaries. Brash and her get along, great, but you're his primary, and from your original post, it seems like she's worked it so that you're excluded. You and Brash are a package deal, she seems to understand that at some level, but she's also not willing to respect your boundaries. Neither is he.
Ok, so Brash is in the throes of
NRE, and yes, you can appreciate it, or celebrate it, whatever. However, he also needs to be aware that his exuberance over the new partner and to maybe tone it down when in front of you. Him not calling you after he's back from a trip with her is a big no-no. Both of them forcing your friendship with her is crossing a boundary that is making you unhappy. I'm seeing a form of emotional blackmail here.
Talk to Brash. Explain to him that this relationship he is cultivating with this woman is straining your relationship. Try to work out a way to resolve this conflict. Then invite the woman over, and the 3 of you talk it out. Maybe she's looking for something he will not be able to provide to her.
But before that, look into yourself. See what it is that is keeping you from liking this woman. Is it an obstacle on your part? Maybe you're not communicating your feelings about this situation properly? If she is so dead set on being your friend, there must be something more than just dating your husband. Are you really giving her a chance?
Please understand I'm not trying to place the blame on you. I'm just trying to cover all the angles, here.
Best of luck, and let me know the turnout.