Re: vee (instead of triad) ... there are cases where the two "arms of the vee" don't get along much and try not to see each other much except in passing. I know of at least one such vee, and I know it can work.
However, when the arms of the vee *can*
have a friendly relationship with each other, I think it's of great benefit to the vee as a whole. My own vee (MFM) has me and the other guy as friends. Not best friends ever, but friends enough to be comfortable with each other. We spend some time together, just us guys, and I think it helps us to stay with the "We're on the same team" mentality.
So, in most cases, I do recommend that the platonic relationship in the vee get its own private time. Whatever those two people enjoy doing together. Add to that the fact that the vee in this thread could conceivably become a triad again later on, and you have my reasons for suggesting the wife and the girlfriend spend some time together. If they just can't do that without driving each other nuts, then I'll modify my opinion.
"In all honesty I think I am trying so hard because I know it is extremely difficult to find someone who wants to be part of this so I feel like she is the only option."
That's kind of a sad thought.