If you are luring/baiting it headfirst into a trap, that sounds like using a cubby or running pole set with a bodygripper which is designed to KILL the animal. So in that case, the animal should already be dead. You are from Maine, and it is legal to use small bodygripper traps that way for ermine, marten, fisher, mink, and raccoon.
The other methods of trapping utilized in north america are snares/cable restraints, foothold traps, and box or cage traps. Use of foothold traps ( aka "steel jaw leghold") on certain animals was employed for research, relocation, and restocking where that specie had been exterpated many years before ( before modern game laws). The list includes lynx, fisher, timber wolf, red wolf, Mexican wolf, coyotes, and river otter. All captured with the trap the animal activists villify to no end, yet these animals were able to be released with minimal injury.
Trapping is the #1 way to capture certain animals- you just can't be lucky and find a fisher and then dart it for a relo prodject. Would you shoot muskrats swimming in water(possible ricochet)? You can trap more raccoons in one night with much much less effort than running your coon hounds around in one night ( though running the hounds can be quite exciting). So if wildlife management is your goal, you choose trapping for certain species. Removing excess amounts of these plentiful to overpopulated animals, helps the habitats and the remaining animals be healthy. Many of them can also be pests to poultry and livestock, and sometimes killing dogs and cats. Coyotes are a major PIA for some farmers and to many suburbanites. Foxes tend to be a poultry killer. There is a marten species that does extensive damage to people's cars in central europe! Raccoons are all around pests, getting into attics and barns and being plain destructive, and going after poultry, as well as being a main killer of wild ducklings and bunnies.
As to dispatch, if you have a live holding trap you can opt to use blunt force trauma to the head, stunning the animal by a blow to the muzzle to render unconcious and then crushing the lungs by standing on the chest ( works very well on fox), or shooting it ( recommended for coyotes, raccoons, badgers, and possums). For wildcats ( bobcat and lynx) you can easily and quickly choke them down with a snare or catchpole.