First off, why the pressure to develop this meaningful bond with your metamours? Now, I absolutely promote that there shouldn't be negative feelings between metamours, but I think being civil is enough. Anything else that develops between metamours organically can be a bonus. By organically, I mean that you might meet at some point and discover that you have lots in common and want to develop a relationship of your own, be that romantic or platonic. But it isn't because you have to, it's because you, of your own accord, met that individual and decided that they were compatible with you, as an individual, for a relationship of some kind. Not because they are your partner's partner and that means you have to be best friends/lovers.
I think what might be causing the problem for you here is that you think you have to have this deep, meaningful relationship with her to make this all work. That isn't the case, the only thing you could do to harm his relationship with her (if he lets you) is be spiteful and malicious by purposely obstructing their relationship. I don't think you intend to do that, so relax with trying to force a friendship/relationship. Let that grow on it's own.