I don't see a looming trainwreck for everyone involved, but I do think it could be disastrous for at least one person (you) and possibly painful for everybody.
Have you thought about just putting the wedding on hold until you two figure out what sort of relationship dynamics you're going to be open to? That is what seems to be more likely to be a trainwreck. I am not sure if you are actually expecting if they are physical, that she says eww yuck that was horrible, lets get married and be monogamous after all.
Is P coming to your wedding? If they start a relationship are you going to enjoy having him at your reception and kissing the bride? Do you want A all distracted by NRE with P on your honeymoon?
I think because it is a whole 3 weeks before they will see each other that it is WORSE while you wait in limbo. Nothing can be settled and you're having hard and emotional discussions regularly while the wedding date looms nearer (is that what you want? are you hoping that if nobody postpones is, A will go ahead and marry you despite the commitment issues you discussed?). Maybe it's no problem to call it off last minute if things go horribly awry, but it seems that things are really too emotionally tumultuous to get married in the middle of not being sure if your relationship is going work out, or if it's going to be non-monogamous.
There's a lot of pressure on both of you with this wedding date - I think you're looking at it as a safeguard (as you said, you're the winner because once she says "I do" your heart is safe?), and she's looking at it a bit as the day the noose goes around her neck, and she doesn't even know for sure if she's running from commitment and being self destructive, or if her subconscious is telling her that marrying you isn't the right choice for her.
Happiness will never come to those who fail to appreciate what they already have.