I don't know of
ANY .22 handguns that can be had for $200
new that I would recommend.
If you look around, you can usually find a
used last-generation S&W .22 auto in good shape in that price range - excellent .22s and vastly underrated IMHO.
But (also IMHO) the very
best used .22 handgun you can buy for around $200 is the
Ruger Single Six convertible. (I bought one LNIB, SS, both cylinders last year for just under $250 out the door; I see unboxed ones in decent shape for around $200 all the time down here).