I think the vet should do the deed; because the medicines are dangerous and should not be handled by lay people. I think if the house call is an option, that's a good way to go, especially if the pet is traumatized by riding in a car.

I'm surprised about how they euth rats. At the pharma company where I work, I think they use gas. Sticking a needle in anywhere can't be very humane, IMO.