Why argue with him? Why listen to him? Why negotiate with him? His word is not bond. He is an active alcoholic. Just about everything he says to or at you or in your direction is empty. Totally bankrupt. Quack quack quack.
I know it's too late to change plans and that you're leaving today, but keep this one fresh in your memory -- that your alcoholic husband is so selfish and cruel and sick that he would keep a suffering animal on the verge of death alive for his own benefit. Not yours, not your son's, and certainly not your dear old pet.
The death of a beloved family pet is harsh under any circumstances. I'm sorry you're going through this.
Perhaps it's time to take some of your power back. Let the alcoholic keep his crazy to himself.