It will cost extra, but why not ask your vet to do a home visit?
If they are not being very sympathetic or helpful, then I'd seriously consider changing practices. A vet should be someone you can trust, respect and talk to, often at times of very great stress.
Alternately, I'd ask your vet about a possible tranquiliser to give him before th journey, do vet visit, (dont' know if this is medically possible by the way) for vax and trim, then get him home and work with behaviourist.
I think you should perhaps get a different vet ! Perhaps ask around locally and see if someone could recommend a vet who may be more help. Is there not a vet within walking distance ? If not then I'm afraid I would be getting him in the car . There are a few vets who post on here so maybe one of them could give you some better advice .
You need to bite the bullet and get him to the vets , they can then do his vaccinations and if necessary sedate him to sort his coat out . Longer term you need to perhaps get a behaviourist to work with you so you can groom him as otherwise you'll be in the same situation again a few months down the line.
In August we got a Bearded collie (had had a Beardie cross before). We went for the working type as just a pet. He was about 12 months and had been kept by breeder for sheep dog but it hadn't worked out. We saw the parents etc and had no major concerns. However when we got him home it quickly became clear he had a few issues. He wouldn't come inside at all and was shivering wreck if forced too. He growled at the kids. Wouldn't get in car. Ran and cowered every time I lifted a brush / rake / stick while gardening.
Then he hurt his paw and we had to force him in car and take him too vets. He shivered and cowered so much she was forced to examine him outside. She didn't vaccinate as he was hurt. He has no vaccinations. This worries me.
Anyway after lots of love care and attention we have made significant progress. However he still won't go in car but will now come in house and just recently allows me to groom more than his back and he's severely matted underneath and behind ears.
So what do I do? He desperately needs vaccinated but won't go in car. Also desperately needs groomed (possibly shaved underneath). I am concerned at cost of getting his coat sorted.
Advice greatly recieved. I admit we are well out of our depth but I wasn't going to return him to somewhere he was mistreated.