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That's baloney. Animal controls and shelters catch strays in traps all the time. Dogs aren't nearly as smart as coyotes!
To the OP: your landlord said you couldn't build any more fenced structures -- but how about a free-standing dog pen? You can buy used 10 x 10 dog kennels quite cheaply off of Craigslist, and you can put them up without damaging the land in any way. Do you think your landlord might agree to that? It would at least keep the chickens safer while you're trying to catch the dog. Just a thought!
As for trapping the dog -- if you have a shelter close by, see if they will loan you a trap. Many of them do have loaners. Then put something really nice and stinky in it, and cross your fingers!