My suggestion in approximately that price range is the Ruger Mark II or III. I vastly prefer the bull barrel. The Ruger 22/45 is excellent too with the bull barrel.
The Ruger SR-22P is a good little 22 pistol. Fits your budget. It's not a hunting gun or serious target gun, but a great plinking gun. I am really pleased with mine. But I already have a Mark II and quite a few other ones. So a plinker is fun to have. One needs to remember that the dominant use of most any 22 handgun is plinking.
I've been eying the Heritage Rough Rider because it's pretty inexpensive
That's one I don't have. If you are thinking long term, I would go with a more expensive Ruger Single Six. You will be happier in the long run and won't want to replace it like you probably will the Rough Rider in a couple years after you really start enjoying shooting 22's.