Impossible to say - all vets are different, with varied experience of any one condition.
I am NOT a vet, so this should NOT be taken as an absolute. I speak as someone who once had a pup with distemper (he WAS vaccinated, by the way),
Distemper presents as a collection of symptoms, which can include runny nose, fever, mucky eyes, vomiting, excessive thirst, being off food, diarroea (sp?), hardening of the pads of the feet and of the skin on the nose and, later, convulsions.
A vet's likelihood to spot ANY condition is partially dependent on how often he comes across it, I suspect. Some of he symptoms above could apply to a number of diseases.
If you are concerned but fear the family won't do as you suggest, call the vet yourself and tell them, anonymously if you prefer. A bit of Googling theur town, finding the name of a vet and then a simple "What vet do you use? Why? Oh, because my friend uses Peter Smith in the High Street, she says they're very good. Oh, I see, you prefer John Jones..." type of comment should be all it takes.
It really is down to the vet to decide what's wrong with this poor pup, who sounds in a pretty rough way. Please don't let anything get in the way of that.