Hi, kinda just going though each of the forums and checking it out. For whatever it's worth to you here's my two cents:

I believe the Minerva would win in a fight based on the scenerio and a few of my personal assumptions (opinion, whichever once you want to call them):

First off I'm not going to look at the specs of the two ships because let's face it. Truthfully either one could destroy the other. If not we could just stop there and be done with it. So the assumption maded here is that it will depend on the ability of the crew and captain.

Analysis of the crew:
Based upon what I've observed of both crews in the series I will assume that the crew of the AA is superior:
Superiority comes in the form of, they've had more experience not just in battle but also working together as a team under difficult situations, are able to take commands and carry them out promptly. Know where they're supposed to be at any given time.
My assumption here is based on the fact that crew of the AA have been through more battle scenerios and actually have more well formed and structured chain of command not only based upon ranking but mutual respect and understanding for eachother.
This is something I've seen lacking in the crew of the Minerva where even the vice captain is found away from his post during battle scenerio, orders are not given or carried out promptly at times. And the respect that's due to the captain is not always displayed as other people have put forth as arguments before.

Analysis of the captains:
Based upon limited knowledge of the history of the capatain of the Minerva I would have to say that the captain of the AA have far more experience in battle. Though in terms of ability I don't know. Both seem to be lacking. (I think I know what most of you are going to say but hey that's just what I think...) The captain of the AA lacks in battle tatics when compared to the captain of the dominion which completely out maneuvered and strategized her. As was the captain of the Minerva by Neo at the beginning of the series and by the ORB fleet. Both display a tendancy to be caught off guard and suffer for it.

The advantage the AA does have over the Minerva in this field are two very cunning individuals with commanding experience in battle. Mwu and Andrew. Both of which have far getter battle experience than either captain, and show a cunning and deviousness the two captains seems to lack. (whether it was because they were evil at the time and though of evil plots at the time I have no clue but either way they can be sneaky little...)

Given this it would seem that the Minerva is at a disadvantage up to this point and truthfully yes they are but now we'll look at the motivation of the two ship captains, which I believe to the be the deciding point

The Captain of the Minerva's only motivation is that she feels ZAFT is right and thus her orders are law. She doesn't have to worry about the consequences so much as long as she stands by the" ZAFT and the chairman are pulling the string and I trust them to do what is right in the end so I'll just do my job to the bet of my abilities"

The Captain of the AA on the other hand has only her morale conscience to guide her actions. She takes no orders, everything she does she must first justify to herself. This can leave her nice and conflicted, at time of decision. Something that almost costed her life in Gundam Seed. She still has difficulty with the concept of taking lives as seen in her actions throughout most of every battle the AA has ever been in in GSD.

This I believe to be the biggest deciding factor. Mindset. The reluctance to fight or take a life during battle time is fatal. If you can't pull the tigger at the last scene, you die. Mwu died once to save them from that mistake and given the hidden theme of "not learning from past mistakes" in GSD I think that the captain of the AA is doomed to repeat the same mistake. Given no outside interference by gundams. I don't think it likely that mwu can die again to save her from the same mistake a second time give this hypothetical situation.

I believe on the other hand that the captain of the Minerva, while still reluctant to fire, is less so and will fire when time comes. (this is of course assuming that someone like mwu or andrew doesn't just go off and pull the trigger before the captain has a chance to hesitate, cause the flip side to that is they probably have someone like rey on the minerva side that'd do the same). I my mind it would be the same reason why Shinn was able to be Kira the first time, because Kira wasn't trying to kill Shinn but Shiin was trying to kill Kira. Even though the difference in ability and Gundam Suits gave Kira the advantage Shinn was able pull the trigger in the end while Kira wasn't. Let's face it, had Kira actually wanted to kill Shinn at that point, no questions asked Shinn would be another smear on the ocean floor.