With the condition that eggs be disallowed from the arena, sure. That's one tactic I've seen in the mansion raids where players bring stacks and stacks of eggs to quickly defeat the mobs which, in turn, also circumvents the difficulty factor (as most mobs are on a 100% catch rate).
Yea. I'd say that bringing eggs to capture mobs from the arena would incur a player a penalty same as if they cheated in the combat portion of the match, in game (the typical penalty for breaking a game rule in this event is losing a "win"). Could also warrant them some other sort of punishment for rule violations if you're stating you don't want it done.
I was also considering removing the ship to ship transporter portals, and allowing the use of ender pearls instead.