I personally don't know what the law is. I doubt there is one specifically regarding crocs and what one can or cannot do on one's property. The laws of Belize are now on-line, so if one is interested in spending some time reading dry law books one could find out.
Of course what the local is or is not frankly isn't pertinent. There are laws on the books against cutting mangroves, for example, but that's routinely done, around San Pedro and all over Belize. Having a law on the books, and having it enforced (or enforced consistently) are two different things.
The two crocodile species in Belize are not endangered in Belize, but on the other hand I agree with Karen that crocs were in Belize long before Americans (and their dogs) were, A good steward of the land would not kill a natural creature, one which plays an important role in the ecology, except as a last resort.
Of course we all act irrationally and kill things we are afraid of -- snakes, spiders and such. I myself killed a small, harmless snake last night that was at my back door trying to come in the house. It frightened my daughter so I stepped on it. That was not a good thing to do.
Crocs in Belize certainly do kill dogs from time to, just as alligators in Florida or Louisiana occasionally kill dogs. It is rare for them to bother humans. I recall one incident a few years back near Belize City were a young boy was swimming and was bitten by an "alligator" (as Belizeans call crocs) but I can't think of another recent instance of that happening.