I have a couple of .22 pocket pistols (an Iver Johnson TP-22 and a Phoenix Arms). The Iver Johnson is quite ammo sensitive - it jams repeatedly on some brands, while working fine with others.
ALL the .22 ammo I buy is cheap stuff (Walmart Remington or Federal, CCI Blazers, Remington Thunderbolts, etc.). I don't think there is really much difference between the "copper" and lead .22 bullets - the copper ones aren't really jacketed, just "washed" with a thin copper coating. That said, I still pick copper washed if the price is the same.
Since .22s are so cheap, I'd suggest trying a variety of brands and picking the one that works best in your gun.
Doug