Our vet recommended a muzzle for Sheba, and actually its brilliant for her with her problems - she gets into trying to take it off and forgets to get so scared!
It really depends just how your dog will react to the whole travelling to, new smells, waiting room, strange people etc of the vet experience. Sheeb goes to pieces.
(unless you're reallly against them) Why not invest in a muzzle and practise with the younger one, get her used to it (eg every time she goes in the car) and walk her to the vets, then into the waiting room, let them talk to her etc - the more exposure in theory, the better she should get and if you have to wait anyway....
Ask your vet how they would go about sedating, injecting etc - just so you are happy with the idea!