A sub-200 dollar semi auto, with the power to whack pigeons, is a fantasy on par with a Liberal politician telling the climate panic people to stop bothering him, so he can go out shooting. Not very likely.
Lots of decent used Airguns out there. Look for a .22 caliber, as close to the speed limit as can be got, and accept that you are getting one shot at a time, unless you up the budget to some pretty substantial levels.
Like, for around a grand, and with a PAL, you can get a precharged pneumatic bolt action or lever action rifle that is at least, a repeater. And that's a pretty bottom end rig, once you spring for enough gear to keep the tanks topped off. And a cheap scope.
A Crosman 2289, is about the best you get on a budget. One shot, a bunch of pumping, reload, and the next shot. Air, at least, is free.
With practice, reloading a spring piston rifle is pretty quick, but it ain't no semi either.
Most of the non PAL rated semis out there, are glorified toys built on a crappy revolver mechanism, or otherwise nowhere near powerful enough for the task to be carried out adequately well.
The few that I am aware of, are either antiques (Crosman 600, as on example, also quite expensive) or were never made in appreciable quantities, as the respective powers in various countries either effectively killed them off, or the actual cost to buy them in the face of much cheaper real firearms did.