I have no first hand knowledge of AK protocols, but I would imagine they operate like most zoos. The animal handlers are usually trained to also be the rapid response team in the event of an escape. Since they work with the animals on a daily basis they are more likely to be successful in recapturing the animal. Most zoos also maintain a means to terminate an escaped animal if it becomes absolutely necessary for guest safety, though I would not be surprised if Disney lets that duty fall to the sheriff's office since their response time would likely be comparable to the RRT. For your consideration, I present an image of a zoo performing a zebra containment drill:
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