What is the neighborhood dog and cat population like? My first thought for in town or heavily populated is trapping. A live trap would spare the "friendlies" while a snare would take care of whatever. However, the idea of letting everyone's Garfield or FeeFee out of a Hav-A-Hart three times a day isn't fun either.
Plan B would be that if I remember correctly you have a lot of space to the rear, a .22lr from an elevated position on your own property would either dispatch or cause death over time, minimal noise, and if it is done without advertising it would be done before anyone outside your backyard knew what was happening. Sniper tactics - keep the muzzle in the house, and don't let anyone see inside your hide. You'd be amazed at how a pocket gopher can see a movement at a bedroom window eight feet above them, but if I stay about six feet back I can pick them off all day long.