Besides the low pellet energy, the trajectory and wind drift would be practically insurmountable.
Just watched a video of a guy doing test targets for rabbit shooting with two high-end pcp rifles. The .177 was flatter shooting, but even sighted in at 30 to maximize the point-blank range, it was waayyyy low at 50. He was using a scope with mil dots and a rangefinder to plot his holds.
A large-bore gun in .30 or .357 cal might have enough energy, but figuring the hold would be just as tough. Nothing in the price range you mentioned would be suitable.
A Ruger American in .17HMR with a cheap scope might come in at $400.
