Good vets are experienced in handling skittish animals. Most bunnies don’t like to be picked up or held. Sometimes it’s easier at the vets though because the bunny gets subdued by the fact that it’s in unknown surroundings and not on it’s home-field. My bunnies cling to me at the vet, they jump from the carrier right into my arms. They certainly don’t jump into my arms at home =)
One of my buns was taken to the vet before he was “tame” – he’d been living outside in the wild when I found him and I kept him quarantined outside so we hadn’t gotten to know each other when I took him to get vaccinated. Two weeks after that he went in to get neutered. He was feral, like. But there were no problems at the vet.
As for the car ride, most buns don’t love car-rides. But not all buns go crazy with fear either. Put 2 (pieces of) fleece blankie in their cage/s and let them get their scent on them, then put it in the carrier/s. Provide a few treats, f ex sprig of parsley, cilantro, mint. Unlike other animals, bunnies shouldn’t fast before surgery.
Then when you get them home t they will probably be a bit groggy from the anesthesia. Stock up on their favorite food so you can entice them to start eating asap. They might not want to eat, although neutering is not that big a deal (the bun is put under for a very brief time, the incisions are small), so most boy buns bounce back quite readily.