I have a small farm that is surrounded by neighborhood. I sort of have the same problem. Though it is not illegal for me to discharge a firearm, and there is enough space and woodland surrounding my property to “cushion” small caliber rounds like .22 or 9mm. My 20 or so next door neighbors freak out when there is gunfire on my place, and I probably would too if the circumstances were reversed. So, I bought a high powered air rifle for pest control. There are some with higher velocity than .22 or 9mm and THEY MAKE ALMOST NO NOISE! I would also recommend a cross bow or compound bow. I can’t use a compound bow because of severe back trouble. HAPPY HUNTING!
Have you thought about the new long bows that are out there? Apparently they have a lot of power and they are uncomplicated. I don’t know how’d they’d be with your back.