At any cost?Then get a FX Royal .22 36FPE.It's a very quiet gun
there's a multitude of rifles that fall into those specifications you listed, quiet is subjective to each person, I don't expect completely silent but backyard friendly is my upper boundry. my fx is the most quietest of the bunch with only the mechanical sound.with that cash limit I would look at a reasonable priced rifle and a compressor and tank.you can narrow it down by wood or syn, walnut or lesser wood stock, length, weight, fps, energy,shot count, regulator.as you narrow down the list you will have a better picture of what's out there that fits into your preference.one thing is you can't go wrong with a BSA, daystate, RAW, Weirauch or fx for fit and finish and quality workmanship on the high end side.on the mid to low price you'll keep the performance but lose some or get less of the mentioned above.
With a hawged out valve, heavy hammer spring, power adjuster and a 31 grain swaged slug I can get 40 fpe out of my .22 Disco. I can still keep it pretty tame with my TKO. That all cost me under 350.00. PCP is probably the way you have to go, you just need to figure out what you really want to do with it, and what you really want to spend. With 4k tops, heck you could buy yourself, me, BobbyHumphery and AHMSA a nice PCP. We would be very proud to own a PCP paid for by you. LOL!