Although it is not stated in the opening post, my guess is that the alligator was not already dying when they took it to the animal control center; he was probably quickly put down once they got him there. What else are they going to do with the five foot long alligator ? He certainly is not an adoptable pet, and even if he had been raised as a pet (which is uncertain , too), if he was aggressive enough to be used as a "guard dog" alligator, then he was most likely aggressive to the animal control people who picked him up.
Since he was an alligator, it is probably surprising that they took him to the animal control compound at all and did not just shoot him out where they found the marijuana stash that the alligator was guarding.