Which would work as a better solution?

Having worked for a vet that did all the spay/neutering for the local Humane Society, I would vote for a county sponsored spay/neuter clinic. Or a mandatory spay/neuter program with an exception of a breeder. Maybe you could also have a registration program at minimal cost to help cut down on abandoned pets. People abandon their pets because they can't afford them or the pet isn't what they thought it would be. If someone does move into your community, you could offer them either program, spay/neuter and/or registration or both. At least the pet will be registered and would help prevent abandonment.