Repel Boarders!

You can deploy it only in an opponent's sortie phase, after dispatch (in the second window or the final one), while they have crew in the mothership battle attacking you. Never in your own turn, in a watch or before dispatch. It is checked then, not again as it resolves: if every attacker has left by then, it still does what it can (802.12). Attackers stay in the battle until the end of the sortie phase, so in the final window it reaches the ones that survived the battle.

A condition, not a cost: nothing is exhausted. When you deploy it you need a Brigantine on board that isn't dispatched, ready or exhausted, new ones too. A crew of yours in a battle doesn't count (306.6).

No battle is chosen: the mothership battle, as the card resolves. Each of your opponent's crew there takes it; your own defenders don't. Plating takes it first, the damage stays, and a crew left with no hull is destroyed once all of them have taken it, before the battle is fought if it is still to come. A crew that has left the battle by then isn't hit (705.7).