Well I'm going to look at this problem solely from the view of keeping the dog from experiencing unnecessary pain.
It's not about him, it's not about your marriage, it's simply about the dog being in pain.
Just put it down and alleviate the suffering, the quicker the better for the dog.
Did your H actually say anything about that keeping the dog alive had anything to do with his sobriety? If not, why are you putting words in his mouth and head?
That your H said he would "kill the dog for you" ...very accusatory and immature...and a guilt trip too.
You could call him up today and tell him you are putting the dog down. At least he would have notice of what you are going to do.