As someone mentioned earlier, I'd try to lure them more towards the backyard to dispatch them if you can, much less risk of involving neighbors/their property with ricochet.
Trapping wise, I use a conibear set up tied to a string and attached to a garden pole with peanuts on the bait triggers, though with your wrist injury I wouldn't recommened it, could be bad if it slams on your fingers trying to set.
I also have a cage trap, not the squirrelnator, one of those smaller ones you can pick up by the handle and could hold at most 1-2 squirrels, and in my experience those work much better for tree squirrels than the big squirrelnator (I've seen squirrelnator better success at ground squirrels than trees). I prefer buying a large bag of peanuts like from Sams Club or some other store, usually cheap and they cant resist them. Take a small lid, like a peanut butter jar lid and put some peanuts in it and put in back of trap. I also put some peanuts in one of these
https://a.co/d/d1AYYAi Just hang it inside just above the plate that closes the cage (sometimes squirrels are too light and dont trip the trap, but when they try grabbing this it makes them push more on the plate). And if you really wanna try to lure them in, scatter some peanuts around toward the trap and some aniseed oil.
Then you can also bait lure them to a safe spot to try and shoot them can be either front or backyard, just make sure you have a good backstop or wall, etc. Get a jar of peanut butter, fill with some peanuts or bird seed, and lay in front of the backstop. Whatever airgun you pick, just practice with it prior to make sure you have it sighted to hit whatever is near the bait. With this, if you dont mind olaying the waiting game, you can leave the bait out and let the squirrels go at it for a day free of charge, this makes them use to going to the spot. Then next time they come is when its time you get rid of them for good hopefully.
As for what airgun to use if you still need one, with your distance to the outside, a Hatsan Supercharger QE or Supertact QE (same gun, just has a stock) in .177 or .22 could work.I've taken several with the supercharger in .177. Might need some force to fully break the barrel and cock with your wrists but dont think its too bad. Theres also the Seneca Dragonfly mk2, always heard those are easy on the wrists when it comes to pumping it.
But if you dont mind spending more, you can get a nice cheap air compressor and have access to a wide variety of PCPs that to deal with all the squirrels and without making your wrists hurt. Tons of good budget ones like the PP750, Notos, Diana Stormrider, etc. I wont recommend any of the higher end ones as those you really wanna be hooked into airguns to get into