If there was a speed penalty, it might be balanced. As Juggler is right now, there wouldn't be one. That the trait requirements are weird for a melee build sorta balances things, maybe. Making a dual wielder trait that unlocks after the melee build traits would be broken, though, especially considering you're getting all the Brute bonuses *again* on the second weapon. In no world do I see one trait giving you the ability to do an extra 2d5+12 damage being anywhere near balanced.
Maybe if you didn't get Brute or Berserker on the second hit and there was a good speed penalty. Maybe.
It only seems unbalanced in the absence of context. Going through your points in order:
Yeah, a speed penalty would be a necessity, so that there's an element of tradeoff as in Dualgunner. The penalty should also be higher than Dualgunner's, since melee weapons are heavier.
Having non-melee trait requirements would make such a trait useless, because by the time you could reasonably unlock it you'd be past the UC. On the other hand, having requirements like Brute(3) or Brute(3) plus Berserker(1) would grant this ability available in one of the most unbalanced portions of the challenge, namely shortly before the Chained Court.
More importantly, in the AoB challenge, doing an extra knife's worth of damage doesn't matter very much once you have the chainsaw. At that point, if you can get yourself next to an enemy, you can kill almost anything in one or two moves. With an extra knife swing, again, you can kill almost anything in one or two moves, given that you've probably skipped a level of EE to get the new ability. Doing massive amounts of damage is _already_ the melee character's strength, and adding more doesn't do anything at all about its severe weaknesses, namely having to run up to an enemy and having to deal with multiple enemies at once.
In fact, calling the new skill DW, my ideal skill progression would look something like this, assuming DW at Brute(3)+Berserker and a speed penalty for it:
Bru->Bru->Ber->EE->Bru->HR->Iro->HR->DW->whatever...
It wouldn't be a particularly compelling skill like Berserker: I would aim for it to be my last skill before the UC, because that's the only place before the CD where the extra damage would make a significant difference.
I'm not sure about your Berserking suggestion. On the one hand, it would decrease your already low chance of becoming Berserk before the AoD. On the other hand, you could unequip the extra knife until you reach him, where the extra damage would certainly help.
Anyway, thanks for the explicit objections. Those are much better than just a statement of preference. :)