OT: need advice, BFF and fiance
babymama22 is right about how you need to sincerely make them acknowledge that between their fighting, you are also getting hurt. Solid, worthwhile relationships require attentiveness and compromise; they need to be mature enough to realize their behavior isn't being geared toward solving what's wrong, and that is putting a lot of stress on you.
If you are as close to your best friend as I am to mine, I could predict that you will almost never be moved to cast her away. Your fiancee is being irresponsible by disrespecting the people you hold close to you. Both are being selfish in the same way: they want to be at odds without realizing it, feeling slighted by the other and wanting to win over them. Instead of solving the problems or at least trying for some middle ground, they're disregarding their ties to you. That's unfair, and you ought to make that clear to each of them.
I don't think seating them together to 'have it all out' will solve anything; in fact, it will escalate things because both of them will have agreed to it grudgingly, which results in the inability to listen (they're just going to want to keep pressing their points against each other).
Before you put your foot down, let go of any frustration and concentrate on how much you love these two people and need them to not necessarily see eye-to-eye, but consider your feelings and just, if nothing else, agree to disagree. Tell them to act respectably instead of insisting on battleship tactics.
I won't sugarcoat it; I don't know the entire situation, but a sensitive girl and an insensitive guy can really drain even the most persuasive efforts. You may be in a situation that will inevitably cause you to fall away from one or both of these people. If it's as bad as it sounds to me, it needs to be cleared up pretty soon or you'll all be too tired to make it right.
Good luck, girl. :)