Before starting, I would like to tell you that I am a fan of catch, spay/neuter release programs, and I applaud both the people and the vets that are dedicated to them.
On a positive note, it is great that you already have some repore with the cats, so there is some level of trust. However, being feral cats, that trust will not go that far as to realistically being able to handle them without very likely getting bitten or scratched. Even more tame ferals that are used to being fed and in the vicinity of people, can act akmin to raccoons when attempted to be handled or restrained in any way.
Therefore, when my clinic was involved quite a bit in this type of program, we had the feline rescue people bring the cats right down in the traps, and we would take it from there. Your vet should be equipped and able to deal with getting the kitty out of the trap if he is used to working with ferals.
What we would generally do is open the front of the trap and pour the kitty into a plexiglass box with a hole on the top that enabled us to pump anesthesia gas into the box and allow the kitty to peacefully nod off. Once asleep, we would place a mask on the kitty if it was a male and therefore simple castration, a tracheal tube if it was a female having a spay. Prior to alteration, however, we would do a quick blood test with in hospital snap kits that would tell us if the cat was positive for FIV or Feline leukemia. If positive, we generally recommended euthanasia, as that was a source for infecting not only other feral cats, but people's house cats that they let outside and may come into contact with the feral later on. If the snap test was negative, we then proceeded with the alteration, gave the kitty a 3 year rabies vaccine, then woke the kitty up in a regular cat crate for the shelter people to pick up, take them back to their territory, then simply open the grate and let them leave.
Other veterinarians with give the cats a sedative injection through the grates of the trap, but I think that is ultimately more stressful the the "cat box" method we favor.
As far as minimizing stress for the kitty in the trap and in transit, the best thing to do is minimize stimulation. I would cover the trap with a porous blanket to alleviate visual stimulation, and try to talk softly and minimize noise as much as possible.