I don't use live traps often. I own 2 havahart traps but I took more critters with a homemade box trap.
With the box traps I used a hog catcher noose to pull them out and then shot them.
I've both shot the critter and drowned them with the havaharts.
The problem with shooting them in the trap is the blood.
Don't really want it on the trap.
With drowning you now have a wet dripping animal to handle.
Now you have to drag it out of the woods.
In the dead of winter they don't dry very fast either.
One method I haven't used is to inject it with something.
They used to sell something that you use in a syringe on the end of a stick to dispatch skunks.Dunno what was used.
Something I've been considering is using dry ice and covering the trap ,critter and a chunk of dry ice with a blanket or plastic. Ought to work but not really anything you'd want to be packing around on the trap line.
I tried using a plastic sheet and spraying auto starter fliud (ether) to dispach a skunk. Man you'd be surprised how long a skunk can hold its breath.
Either that or I didn't use enough or wait long enough.
Lotsa luck
just