In most places in the US, relocating wildlife, particularly predators, is illegal. With that in mind, live trapping may not be the best method. It stinks, but it is, unfortunately, the law.It will help to visit the neighbors to alert them, and anyone within about 1/2 mile who obviously has chickens or other birds. Then everyone will be on alert, and maybe someone else can get a successful shot at him.
There's information about live trapping foxes, so add that to your plan.
Meanwhile, keep those birds safely inside!
Good luck,
Mary