I've never used them but some folks report success with Rescue Remedy or there is a DAP for cats. Maybe someone here will talk about them, or you can google. Friends with a cat like Poppy avoided the Vet for all but the most dire reasons and then they had to sedate the cat first, sorry I don't know how they did that but I think it was a pill.
We are lucky, our feral just freezes at the Vet's, she's too terrified to move. Except after a stay for an undiagnosed illness when she was to go back for final bloodwork. Not nice of me but after years of me warning them about this cat she'd finally had enough and went straight up the Vet wall, somehow clung to the top near the ceiling then had to be fished out from under a piece of furniture. They gave her back to me saying they were going to skip the bloodwork, she seemed to be quite recovered. Perverse of me but I couldn't help thinking, "see, I told you."