The PCP guns are 'sexy' and the hot thing. I get that. But a springer will be less expensive. It won't not need to be 'pumped' up 100 to 300 times. It won't need a refill tank. You don't have to worry about the aluminum air tube corroding due to moisture. It's just less maintenance and trouble, all at a lower cost.
Noise-wise, the sound is going to be about the same without a suppressor of some sort. Just stay sub-sonic with a .22 or .25 cal gun. Don't get a .177 going supersonic, or noise WILL be an issue.
I do think the .22 cal is the best choice. The .25 would work great, too, but you can find .22 pellets everywhere and having shot bunnies with .177 and .22 pellet guns, I would never go back to using the .177. The .22 was more 'authoritative' in putting the cottontails down. I'd love to try a .25 on 'em! But for the OPs needs, it'll be louder, a richochet will carry farther, and finding pellets will be trickier (not going to be at Walmart.)
I have an old Webley Vulcan springer (1980s vintage) in .22 cal that puts the smack-down on rabbits and squirrels. It makes a 'crack' sound when fired, but the neighbors only hear it once (rabbit is dead) and they just think I hit something with a hammer or dropped something loud on the ground. Keep a battery powered nail gun (Ryobi makes a nice 18v unit) with a 2X4 board with a nail in it by the back door. If questioned about 'noise', just produce the nail gun and board and say, "sorry...working on a project" to explain the sound.
Honestly, unless the neighbors see me standing there with a 'rifle' in my hand, they think nothing of the small noise the springer .22 makes when dealing with my problem vermin.
I will make this observation though - I cannot scope my 'old' springer. It has destroyed two scopes and the one that survived would scootch along the mount no matter what rings I used or how hard I'd clamp it on there. Newer spring piston air rifles address these now "known" issues with better bases and rings and there are scopes made to survive the unusual recoil a springer produces. Whatever - I've been using the iron sights and doing just fine out to 40 yards for 39 years with them. A scope isn't really needed for suburban 'backyard' distances.