Go with your gut. A lot of times the dog is having trouble because we are concerned and they sense that. I could take my parents' dog to the vet no problem, but when my dad took him he huckered down with his tail between his leg. He LIVED with my mom and dad. He trusted them fully. But my dad was raised Christian Science and was raised to be terrified of the medical profession. My dad had six kids and he took them to hospitals and doctors as much as my mom, hiding his feelings about it, so we wouldn't be stressed. But you can't lie to the dog. The dog reads body language and any associated smell that happens with fear or other strong emotions -- not silly, obedience competitors will pop a mint so as not to throw their dogs off by their fear, through their breath.
So, maybe, maybe not. My dogs will allow people to take them out of kennels. I do not muzzle them unless I know there is something very painful/scary to be done to them, and then I remove the muzzle immediately, but, my dogs are very unlikely to bite in any circumstance. So far I've done both, I have stayed throughout, and I have left dogs at the vet to be anesthetized, etc. Dogs and humans were fine. Again, you have to go with your gut because you know your dog. And maybe, you need to trust your vet. Chances are pretty good that they have managed dogs that are more scared, more aggressive than yours without a problem.
Jenna, RN CGC & Babs, CD RA CGC HIC
Heidi, RA CGC
SG3 Odessa, SchH1, Kkl1, AD
Ninja, RN CGC & Milla, RN CGC
Joy, Star Puppy, RN CGC
Dolly CGC & Bear CGC