A ten day quarantine period is standard for possible rabies exposure in a lot of places. Rabies can be passed to humans and other animals, which is why they want the cat kept away from you and your other pets. That all sounds normal, although I'm sure it's very stressful on you.
What puzzles me is that you do have proof of vaccination. As far as I know in my area, the quarantine is for either unvaccinated cats or cats whose owners can't prove whether or not they are vaccinated, not for cats who have their shots.
It may be different in North Carolina, but have you specifically told Animal Control that you can prove your pet is vaccinated against rabies? I know your vet knows, but did you mention it to AC? It may not make a difference where you live, but here it would.
I don't know about enforcement in your area, but here they definitely do enforce it, so I wouldn't take chances. I would do what they say. However, I would find another vet I think.
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The Boys: Maisie, Zephyr, and Cullen.