As stated, most people kill predators including myself. Relocating often results in the animal finding its way back to its territory or else just putting the problem on somebody else. It sounds like a raccoon or an opposum killed your duck.
I would set out the traps, but be warned that I used some of the same and while I caught an opposum, the raccoon I trapped was able to escape from the trap within 20 minutes. He was then trying to break into the coop when I shot him.
You want to catch it but you need to do something to prevent this from happening. Lining the inside 3' of the area with 1/4" metal hardware cloth will prevent preds from grabbing them. Also, a secure nighttime house will solve many problems.
I am no longer in the suburbs. But I used to work at the SPCA in FL and we would have people drop off raccoons, opposums and such. Opposums, we'd rehab but raccoons were euthanized. It was state law in Florida. I would check with a county extension office. While rabies is not at all common these days, areas where there have been cases usually don't allow for the release of potential carriers. If you don't have any luck with shelters or animal control, call a nuisance animal remover. They may charge you for picking up the animal but they may also be able to tell you what your options are.
I forgot to add, I wasn't able to kill an opposum with one shot, point blank with a hollow point 22 so I don't think a BB gun will help. Drowning is a slow way to go too. I will kill whatever is going after my birds but I still try to be quick and as humane as possible.