The simple, cheap, most effective answer is the Crossman pump. You don't have to kill them to clear your yard. Three pumps will sting their tails at 25 yards. They will avoid your yard for a good while. BBs are cheap.
Don't bother with a scope. Wouldn't try to hit them at a greater distance due to the possibility that the BB will carry into someone's yard.
If anyone asks, you are just simply stinging them into another part of the neighborhood.
Woodpeckers are a different story.
Others here have suggested the same thing; just a few pumps on a decent BB gun is enough to scatter the little rat bastids, and HIGHLY unlikely to do any damage to the fence, neighbors, etc., etc.
To be brutally honest, I would really rather NOT kill the little beasties.....just chase ‘em off, it that would work.
So appreciate your post; has definitely changed my thinking.