The key might be to advertise in other sections.

Not that many people come to the arena for the reasons you all described, but one good thing to try might be making a setting, and then advertising a tournament there in "General interest checks" or somewhere that gets a bit more traffic than the arena.  People would be able to RP in the tourney thread within the setting, and also fight each other... but losing wouldn't mean getting kicked out of the RP, only not going on for more matches.

I was in a thread like that on another site a while back (it wasn't in that forum's 'arena' section, but was still effectively a fighting tournament), and it ended up attracting loads of people who'd never fought before- including me.  The thing ended up being hugely popular, and our characters spend as much time dicking around and bantering as they did fighting.  Some of the final matches ended up being really intense, because the characters involved had become close friends IC, and yet were forced to go against each other for the sake of their goals.  

So maybe if we tried kicking off such a thread and advertising it to people in the other sections... IDK.  It would take a lot of time to manage something like that, and I don't have much to spare right now.  Maybe sometime later, though... a forum I post on looks like it may be dying, and if it does kick the bucket, sad as that would be, I'll have more breathing room to maybe start working on something like this.