Bring references from your previous landlords and neighbors, as well as from your dog's trainer. Invite the landlord to see your present home to show him that the dog has not damaged the property nor been a nuisance to the neighbors. Use a dog crate. Landlords are much more receptive to dogs that will be crated when their owners aren't home. Don't think you're being unfair to your dog by moving into a smaller place than what he's used to.
Dogs are very adaptable, they can often adjust even faster than people.
Where he lives isn't as important to him as who he lives with. He wants to be with you, and he doesn't care where that is. In difficult times, people often have to move in with relatives or friends who don't like dogs. This doesn't have to be an impossible situation. Use a dog crate when you're not home or when your family doesn't want your dog underfoot.