Guys, this licensing and drug logging is a cop out of the hypothetical.
I assumed, the OP question was meant in terms of an Anastesiologists ability to do the deed.
Lets be realistic, I can just say, as an Anastesiologists, I will take the dog out back and shoot it. That does nothing for the merits of the argument (only reinforces the stereotype that anastesiologists are good shots), yet I am certain that an Anastesiologists (Damn, I hate spelling that word) is capable of doing just that.
Hypotheticals are only fun when you avoid cop outs and legalities and discuss the issue.
The anesthesiologist is risking their license... Doesn't seem like a great position to put a friend or relative in.
An anesthesiologist can't loose their license for Euthanising an animal. Euthanasia, by definition, can't be 'animal cruelty', as the name implies 'good death'. So, unless the anesthesiologist did something illegal, like robbed a liquor store on their way to euthanising the animal, I would say they were in the clear on this one.
Sorry, I am on edge. Send me my letter already