If you cannot shoot it, and if you live where a richochet can hit a kid I do not recommend shooting it unless you really know what you're doing and are willing to live with the consequences, I find drowning the best thing to do. The OP has already clearly stated that animal control will not take it even if he delivers it. The other methods may work if you are careful.
My concern in this is that no human gets hurt. To me, that is more important than the raccoon. I've seen raccoons up real close. I would never attempt to cut its throat or anything like that.
Drowning is a standard way of dispatching an animal if you want the fur. It does not get damaged that way and is a recommended method.
If you do kill it, which I would, you still have to dispose of the body. A lot of times, the trash or garbage hauler will not take it. The options I have are to either bury it or drag it away from human habitation where the vultures, coyotes, and other scavengers will dispose of it. If you do bury it, you probably should put something over it so critters like dogs, raccoons, coyotes, whatever cannot dig it up. I use some wire fencing weighted down by cinder blocks or pavers.
If something is killing your chickens, you need to do what you need to do. I recently had to kill two very large male unneutered dogs that killed several of my chickens and then came back for more. Somebody dropped them off in the country and gave me their problem. I did not enjoy it at all. Matter of fact, I was very upset at having to do it for the dogs sake as much as for the chickens. I just did not see any other choice in those circumstances. I have taken dogs dropped off in this area to the shelter when I have that option. You try to do what you deem is right, but you do what you have to do.