I've actually been in an 'almost' similar situation a few years ago with a rescue Presa who suffered horrific epilepsy.
We rescued her from a really terrible situation, putting ourselves at great risk from drug dealers, we treated her for over 8 months and as her epilsepsy got worse and worse even though she was on the heaviest meds for her size, weight and age we came to the decision to euthanise.....this was made on the facts that the treatment was not working and Pheonix had started to lose interest in life and also, unfortunately, that our finances were becoming too stretched.
Our vet gave us the option of signing her over to them, they would try to treat for as long as was viable, or we have her PTS........we discussed our financial position with both the owners of the practice and thankfully they allowed us to pay gradually with the full knowledge that we would still be paying after it became the correct time to let Pheonix go!
We'd been to the bank to ask for a loan but they refused......at least we tried!
We got to keep her for a good few months after that but we were paying our vet bill off for over a year after she passed...............That is what you do for a dog, or any other pet, that you take on and love!
When you have a good relationship with your vet they are the best people to help and most will not allow a dog to be PTS without good reason!
I do agree the vet/assistant could have made the owner aware that the dog would be treated after being signed over but some owners, as I've said, are pretty selfish and won't sign over an animal in these curcumstances. I think that whoever did take the dog 'read the situation' and did what they thought best at the time. If the dog was in such a bad state he/she wouldn't have had that long to make their decision.......I'm afraid I'd have done the same thing!