Well, if the challenger killed their quarry for the beast instead of subduing them into submission, I would say they lose their "first chance" at obtaining the bijū. Instead, it should shift to the next person in the challenge list as the first contender for the the freed beast. Even then, that person must now fight a bijū entirely.
That is the punishment for killing instead of subduing. Even canonically, the Akatsuki ensured they only beat the jinchūriki into submission. I propose that stays the same.
Of course, then that system could therein be abused because if the jinc didn't like the challenger, (s)he could easily "allow" themselves to die just to deny the challenger their chance at the bijū.
In which case, after I thought of that point, I might fall back on what I was saying and simply allow the challenger to take the first grabs at an attempt to nab the beast.
As for conditions, just respawns in the same area where the person was killed after a grand reappearance to let others know it exists and is free again after a week to regain its strength.
(It would be interesting to make people wait for weeks depending on tails but no one would follow that. Just a side thought that carries no real value.)