As for how the law views different kinds of guns:
If your mental problem triggered a federal "prohibited person" classification, then you cannot own any modern firearm in any caliber or any fixed ammunition. Not a single cartirdge. Not even at home, not even securely locked in a safe (at least not any safe that you have the key or combination to!) The federal laws for prohibited persons do not apply to black powder firearms that do not use fixed ammunition (cartridges). Muzzle- loading rifles or shotguns, and black powder cap and ball revolvers, circa 1850s, would still be legal.
Airguns are not considered firearms for the purposes of any federal gun control laws. So the pellet gun is kosher for Uncle Sam.
If you are not a federal prohibited person, but yet your circumstances did trigger a revocation of your Georgia weapons carry license, then you might consider what type of weapons that you could own but not carry, or what type of weapons that you could carry because they do not require a carry permit! In other words, in plain English, consider that are some weapons that don't meet the statutory definition of a "weapon" under code section 16-11- 125.1
.
A handgun in 17 rimfire caliber, which fires a projectile that is only 4.4 mm wide, should be legal to carry without a GWL. Because the law on "weapons" exempts handguns that fire a single shot of a size less than .46 cm which is 4.6 mm.
I think the legislature meant only to allow BB and pellet guns that fire .177 caliber lead pellets or steel BBs. However, in their usual incompetence at drafting legislation, or understanding how guns work, the General Assembly instead gave us this crap law which apparently creates a loophole for .17 rimfire handguns.