
I took a very small Chi pup for the first vaccination and the usual health checks with my vet some time ago. They booked the 2nd jab with a veterinary nurse. I couldn't believe it when I felt the stuff run down my hand! She had stuck the needle straight through the skin. No apology, and no mention of not getting it right either!
I was in with another dog soon after, and asked the head vet how much vaccine was needed. When I told him that well over half had dripped on me, he immediately booked the head nurse to re-do the jab. I understand that, shortly after, she gave a talk to all the nurses at the practise, explaining what to do if you get it wrong!
I don't agree your vets argument that it "seeped out" and would make it very plain that I wanted it redone at no charge.