a heavy blow to the base of the skull is an instant death. or you could drown ‘em, but be careful they'll fight hard and most likely destroy your live trap if that's what you're using. pellet gun won’t do a thing on raccoons. if it does break through their thick skin and fat, it certainly won’t do much damage. will just piss them off. been there done that lol.
breaking the neck will also do the trick - place stick along back of the neck (use thick gloves) step on ends of stick pinning 'em down, grab the back legs and pull. takes strength but will dislocate neck and kill instantly. you could always get a choke poll, too.
the coon will come back if you take it less than 20 miles and release, learned this the hard way! your property is their territory now they’ll come back. relocation isn’t humane, either, and oftentimes illegal.