Nope. Expect to get doubles the more you run the arenas. I still haven't refarmed all my Master's and Maelstrom weapons since they made a perfected version bc I keep getting the same weapons, so I just gave up. Plus, the weaps are backbar for me, so there's no DPS decrease if I continue using the old (now normal) vMA weapons.
Having done hundreds of battlegrounds I'm pretty certain this is correct.Not only nope to your question, but (last I heard, anyway) the drops from BGs unfortunately don't work that way either. You do have a chance to roll dupe style pages -- the game simply doesn't award them to you when they roll. So the more pages you collect, the less and less often you'll get a drop at all. So the drops aren't as smart as they seem.
I hope I'm wrong; my info is from the Morrowind/Summerset era.