Yes, there was a lot of miscommunication here... and this should all have been nailed down, with specifics, in written texts...
IMHO, I would have never made the assumption that somebody was just making a donation.
I almost find that assumption to be laughable.
The OP did not send money to the rescue.
She paid specific expenses, for a very specific animal, with the intention of taking that animal home.
Don't some rescues make adoptions pending on a home trial to insure that the home is suitable for the animal. Isn't a trial period almost to be expected.
I almost wonder if all the hemming and hawing and 'waiting' was a ploy, and obstruction.
There is no way that vet should have even held back on letting this woman take this animal home.
There is no way they should have pushed to adopt an animal, or take an animal, who has a 'responsible party', waiting to take them home.
THAT might be the only real 'ethics violation', involving the Vet.
If holding the animal for a few extra days were necessary, that should have been an agreed upon expense.
Which would have continued to establish the OP as the owner/responsible party.
The OP's name should have been associated with the animal from the very beginning. The OP should not have been the one who paid with their personal credit card, if they were not a responsible party. And the person from the rescue does not seem to have done that.
XXXXXXXXX is the responsible party (not the rescue) and XXXXXX will pay for these estimated expenses.
OP... Why did the person who took the animal to the vet not list you as the responsible party.
This is where all of this started to go wrong.
The OP has learned some hard lessons here.
YES, what happened was not the right thing.
But, sadly, I don't think there is enough here to back up any official or legal recourse.
Anything involving this amount of money, or an animal that one might consider as their own and become attached to, should be specified, somehow, in writing.