Since your vet is tied up, but not too worried about Smoothie, I would absolutely tell them I was going to drop her off before their regular office hours on the first day they are open. Tell them you have to drop her off because you can't get off work. Tell them you aren't staying to see the vet, but you want her seen. When you get there knock. The staff should be there to take her in. They seem to want to see her, so this should work for you and for them.