Your price point is the sweet spot for at least 4 manufacturers. The family and I love .22LR. We have had five pistols (currently shooting Brownings)
Browning Buckmarks (campers especially) are in that range. They can be
extremely accurate and are easy to take down and clean. We have two of them and as a family love them.
Ruger has a .22 (also at Academy). It is more difficult to clean but is very accurate. We have one of these as well that I still shoot. It has a very wide, easy shooting trigger. If you can spring a little more, the target versions are truly sweet.
Smith and Wesson. I don't have one, but have heard that the cleaning and accuracy of the pistol are excellent.
Beretta Neo. A little different as the grip angle is more radical. Cleaning is hyper easy on this one and it is quite accurate. I found the Ruger/Browning to be more accurate but was able to put ten into a nickel sized target at 8 yards. Absent the mechanical difficulties (which were several) I absolutely loved this pistol.
1. Try them out in the hand. Grip feel is very important and is a big driver in Ruger vs. Browning vs. Neo fans. All of them are going to be more accurate than you are. Additionally all the ones noted have excellent adjustable sights.
2. All my .22s (NEO, RUGER, Browning) had mechancial difficulties and breakages. Browning customer service was excellent. Ruger's was good as well (we literally shot a target version to pieces after about ten years and 50,000 rounds no joke). Neo we had to send back to OEM (Beretta) and they did not properly fix the problem that it came with. Based on that I would not recommend a NEO. Your mileage may vary.
I have not had a S&W in .22 but their company reputation for CS is very high (full disclosure I have three S&Ws in other calibers) and back when dinosaurs roamed the earth and I had a a problem the item was fixed quickly and for free.