You will pay more for a quality air rifle than a firearm for similar power levels.
If you don't even want to spend what a decent .22lr costs on an airgun you are not going to get what I would consider good quality.
The mentioned 392 can meet the need at a low end but still costs almost $200 when you factor cost, shipping, and a decent quantity of .22 pellets which cost more than the more common .177.
It has a few limitations, brass instead of steel barrel being one draw back, but one I have not seen any problems as a result of using lead pellets. They put out an average 14-16 grain 22 pellet around 500-600 fps which will drop rabbit sized game humanely.
A .177 gun needs about twice that velocity to reliably and humanely take something rabbit sized.
You can build up or modify guns from existing platforms to have more power on a budget, but it takes some understanding of the various weak points and how to compensate for them.
There is a number of inexpensive break barrel guns out there. I personally don't like break barrel designs. They are popular because of the convenience of a single action to get the gun ready to fire again. However they have a piston flying around at high speed in the gun that gives the recoil characteristics that can damage optics.
With a pump you would actually likely be just fine using an inexpensive rifle scope from a big box store because rather than send a piston flying that then abruptly stops generating recoil in both directions they merely pop open a valve and release compressed air. So pump guns give recoil characteristics similar to a rifle, while spring piston guns like virtually all break barrels do not and damage scopes not specifically made for them.
Cheaper break barrels are also unlikely to keep the barrel returning to the exact same spot each time, and that means any type of scope mount behind the pivot point becomes gimicky. The level of reduced accuracy means you would be better just using the ironsights attached to the barrel to begin with. Which is fine, as long as you don't have expectations of longer range hunting with a scope.
Finally diabolo pellets bleed energy rapidly because of thier high drag design which is what partially stabilizes them. They lose energy much faster than bullets of similar weight or size.
This means what is a gun of plenty of power for your needs at 20 yards may not meet them at 50 with very different levels of remaining energy. As a result you need what is an overpowered gun at close range to be just adequate at not that much additional distance.
Pellet guns putting out either .177 or .22 projectiles at .22lr velocities (but lighter) are also similar in noise level to a .22lr firearm.
So with typical designs especially those in the lower price range don't expect to be able to plink or hunt and draw any less attention than you would with a .22lr firearm.
