And another look was taken... it would appear I was Very close to begin with all of that time ago, in this small section of data there are 5 checks on the Avenger Sword, but as far as I could make out the game doesn't look at the majority of them and this threw me off quite severely.
However I believe I have finally pinpointed the "get rid of menu" function of the Avenger Sword and have found the means to move it to any given weapon. This coincidentally enough fixes Square's little glitch of not having the Avenger Sword's attack power taken into account. When the Avenger Sword "get rid of menu" part is changed to the Shadow Sword, the Shadow Sword will not save the damage of the last sword it will use its own attack power as a Berserk weapon. I really now have to wonder if that is a glitch or if it was intended to work that way for some reason... In any case here are all of the checks, but I could only figure out what one of them did and that's the first one which gets rid of the menu and gives the character Berserk status... seemingly.
Looking at the data I see now that the game makes a Continuous check to this 3A402B value every second or so to see if it is the Avenger, if it is not it will move through several more checks to see that it is Truly not the Avenger oddly enough. These checks go wholly unused, because the Actual Avenger jumps forward in data past All other checks while other weapons go through these continual checks but oddly the value of the weapon becomes a non-matter before it reaches Any Other Check. It is really quite bizarre and screams of possibly sloppy programming (or possibly just sloppy code reading on my part, heh)
I'll list these highly redundant checks just in case someone can figure out if they have an actual function.
In RAM at...
03A42B - C9 4C - CMP 4C - Seems to be responsible for the menu-cancelling and loss of control of the Equipped Character. Change 4C to the desired weapon to make into the new Avenger. With just this check alone the game treats the new weapon Properly and doesn't ignore its weapon power. There Square. I fixed your game (If it's a glitch). The chosen weapon will jump past these extraneous checks and go to 03A4D8, which is close to the status to inflict on the wielder, unfortunately even if the status is not Berserk you lose control of the character and they will be made into a Proto-Mime.
03A432 - C9 4C - Left Hand (As above)
03A4BD - C9 4C- CMP 4C - Is the "Start of Battle Avenger Sword". Change this to your chosen other value to make sure the Avenger Sword doesn't become a Berserk Weapon again.
03A4C5 - C9 4C - Left Hand (As above)
03A4DD - 09 04 - Loads 04 into 7E2204 - I would record those addresses, but by not being able to take control of the wielder I can't see the benefit of listing those for the lack of current potential use.
In ROM at...
0x01A62B - - C9 4C - CMP 4C - Seems to be responsible for the menu-cancelling and loss of control of the Equipped Character. Change 4C to the desired weapon to make into the new Avenger. With just this check alone the game treats the new weapon Properly and doesn't ignore its weapon power. There Square. I fixed your game (If it's a glitch). The chosen weapon will jump past these extraneous checks and go to 03A4D8, which is close to the status to inflict on the wielder, unfortunately even if the status is not Berserk you lose control of the character and they will be made into a Proto-Mime. I think they had larger plans for this than just being a Berserk Sword, that didn't pan out.
0x01A632 - C9 4C - Left Hand (As above)
0x01A6BD - C9 4C- CMP 4C - Is the "Start of Battle Avenger Sword". Change this to your chosen other value to make sure the Avenger Sword doesn't become a Berserk Weapon again.
0x01A6C5 - C9 4C - Left Hand (As above)
0x01A6DD - 09 04 - Loads 04 into 7E2204 - Berserk Status - I would record those addresses, but by not being able to take control of the wielder I can't see the benefit of listing those for the lack of current potential use.

Looks like one of those checks aren't so redundant after all. For whatever reason the "Equipped During Battle" and "Start Battle With" CMP's are two different things. In this case it is 03A4BD, which is the "Start of Battle Avenger Sword".
I had a theory that one of those other checks were for falling with the Avenger Sword and being revived, but that doesn't appear to be the case. To be safe it may just be best to switch all of these 4C's to your chosen weapon to assure problems don't pop up. (Though one of these CMP's may ironically be responsible for the weapon power glitch of the Avenger Sword.)

2 Some further testing shows that it is only Weapons that can be afflicted with this.
Oh! Hands of Course! I was just testing this with an endgame party and though the Avenger didn't affect Cecil anymore it still affected Kain. I was really confused for a minute there until I realized that those other checks are almost Certainly in regards to the Left Hand.
And that solves the Avenger Sword...! Except for the loss of menu. Also none of those checks turned out to be redundant after all but are in regards to Left and Right Hands as I had seen in the Command Requirements that are explained in detail in the Unknown Info. thread. Meaning you could have Kain go Berserk with the weapon, but have Cecil use it like a normal blade or vice versa.
In essence you can have four "Berserk" weapons. The only issue would be that each one would have to correspond to Hand or Start of Battle.