You will not earn Pet Event basket rewards (Pet Food, Jewels) or standard battle rewards (Gold, Souls, etc) if the Pet Event ends during a battle.
All baskets or stage rewards for battles fought will be sent in the mail (such as Jewels, Pet Food, and Gold) if the Event ends and they are not collected. However, Pet you are trying to rescue will not be awarded unless the final battle is completed before the Event is over, even if you finish the battle within a matter of seconds.
