I pondered this thread a long time before replying.
I am a dog owner/trainer. Every dog I've had the privilege of owning and spending time with has been a valued member of the family and part of my life.
I take the full gamut of ownership seriously. I've spent thousands of hours and dollars over the course of several dog's lives, to treat them well, take care of their physical needs. The ribbons and certificates and titles don't mean anywhere near as much as the quality time I've had giving them a little fun in the bargain, and developing a unique bond.
Now I know that there are those for whom dogs are "tools" who serve a working need on the ranch or the farm. And some of these dogs never become more than that...and that's ok.
I also know that some have had to do the sad chore of euthanizing due to logisitics and geographical need.
[Side point...none of my dogs have ever feared the vet's office...and, depending on need, I have four that I go to. Each vet would volunteer to euthanize my dogs for free if I couldn't afford it.]
All I can say is this...if an owner shoots a dog [illegal in many areas, not an unlawful firearm discharge...it's considered animal cruelty by law] the shooter should be prepared for the gore, the chance he might miss, probably some guilt afterward.
I know that drug euthanization is painless and dignified. I put my best labrador friend down at three due to cancer...and would never have considered shooting him myself.
You've been put in an awful spot. I also agree with previous comments that this dog may never have been properly cared for.
If I were in your shoes, I'd talk some vets. See what they say.
And if you wind up having to shoot the dog because no one will help you with alternatives...I'd insist that the owners come out with you, dig the grave, and then let them walk away, but remain in earshot. Maybe the dues they pay in that way will remind them of their responsibility next time they want to take a dog into their home.
Good luck, I trust you'll find a good answer to this problem.