There are a multitude of .22s that will fill your needs. I own several different rimfires and they are all different in some way. My (arguably) most accurate
.22 is probably my Belgum Browning Gold Medalist Olympic. It's a better pistol than I am a shot, but they are pricey. My son has a Ruger Mk III target that has a scope, target grips, and also has had professional trigger work done and will hang with the Browning shot for shot. But again it is out of your price range. Honestly my biggest suprise of last year (gun wise) was a S&W .22 A with 7" barrel that I picked up used in new condition with box and papers at a Pawn shop locally for $199.00. I have shot it about 200 times and it's accuracy far excedes it's price tag. I thing SRP is about $325.00 or so but deals are out there. Just my .02 cents.