I'm so very sorry. I understand exactly your reasoning, since I just made the same decisions.
FWIW, I've said goodbye at the clinic. It isn't the warmest environment, and it doesn't allow the other animals to see the remains, but at least I can leave and let the clinic staff handle the post-mortem details. Perhaps I would have decided differently if my vet had offered to come here, but she didn't and I didn't feel the need to ask.
ETA: Moving the body would be a major concern here: too many stairs, a not-very-tall vet with a small car, and so on. Plus I've found that I want to leave as soon as it's over.