What a great question and a tough one. I do not have a final answer to this but i will try to give my input from a point of view of Theravada buddhism.
If i were to answer your question "hardcore" i would say that a Livelihood where one kills other living beings cannot be considered Right Livelihood.
Why is that?
Let's take a look at the 1st precept and which factors one must fulfill in order to break the precept.
- Panatipata veramani sikkhapadam samadiyami; I undertake the precept to refrain from destroying living creatures.
In order for one to break the first precept the following factors must be fulfilled;
- pano -- presence of a living being.
- panasannita -- one knows that it is a living being.
- vadhacittam -- the intention to kill.
- upakkamo -- the effort to kill.
- tena maranam -- the resulting death of that being.
If these factors have been fulfilled the first precept has been broken.
What if the vet kills the animal out of compassion?
From a theravada buddhist point of view it is not possible to give rise to the intention to kill when certain mental states are present, for example compassion. When these states are well developed from following the noble 8-fold path then the intention to kill cannot arise since it is an unwholesome state of mind.
In that sense i do not think the vet does it out of compassion. If he had compassion he would not give rise to the intention to kill the being. He might give rise to the intention kill out of ignorance of the true nature of reality. Because of the ignorance he might believe that he is helping the animal by killing it.
What is really going on here is that by seeing that animal suffer, he is experiencing the 5 aggregates. I would think that he is especially experiencing the 2th aggregate of sensations. When viewing this from the perspective of dependent origination (paticca-sammupada), ignorance of the true nature of reality, founded by not understanding the 3 signs of existence and the 5 hindrances, is what leads him to create kamma.
If we jump to the 6th link in the dependent origination which is contact we will see that when the mind goes out to the object and contact is established, there arises feeling. Feeling can be pleasant, unpleasant or neutral as the initial reaction to contact.
I suggest the feeling the vet experiences here is unpleasant which leads to aversion. That aversion is what later leads him to create an action.
On basis of the 3 signs of existence, namely anatta the vet is here creating a self in the 10th link of dependent origination which is becomming. According to the Buddha one creates a self in 4 ways. Let's take the 2th aggreate as an example.
- I am the feelings
- The feelings are in me
- I am in the feelings
- The feelings belong to me
What is happening here is that the vet is creating a self. In order to contain the feelings (the object) and react to them one needs a vessel, a container. That container is the self that is being created. The vet is now identifiying with the feelings. He is taking ownership of them. In reality the feelings are just arisen impersonal phenomena that will cease to exist again.
The feelings are not his and do not belong to him.
Because he has no control over them. He cannot say "i want to feel only positive feelings and never negative feelings". That is impossible and because of that the feelings are uncontrollable, they are ungovernable. What is uncontrollable and ungovernable is oppressing and that is not fit to be called a self.
What the vet should do in that situation according to buddhist practice is to merely observe the phenomena that has arisen and then observe how they will disintegrate after a certain amount of time. What is subject to arising is subject to cessation.
In other words - do Vipassana meditation and learn about the true nature of reality. He will discover that a self is nowhere to be found. Everything works just fine without the need for a self to exist.
Now i would like to make another important point regarding the interfering with the death process of a being.
The fact that the vet does not know what the experience of the animal is the vet might interfere with the death process of that being.
He might interfere with that being's comming to terms with its own death. We know that death holds some of the most profound insights that can have great significance for a beings path to enlightenment. By ending that beings life before the death process is finished the vet might take away that beings possibility to achieve these profound insights.
Also that being is experiencing the results of its own past deeds so by killing it in the middle of that process the vet breaks that “kamma-receiving process” which might result in that being getting a unfortunate rebirth where it must continue to receive the unfinished kammic results from its past life.
Suffering is like the fertiliser for our spiritual life. Suffering should not be "stopped" by killing (reacting) to it. Instead it must be understood in order to achieve insights based on it.
I think that part of the problem is due to most beings spending most time in conventional reality (sammuti sacca) which is based on concepts for example "a man, woman, animal etc.". These things do not really exist. They are merely convenient designations we apply to the comming together/a collection of a number of factors.
This misunderstanding of reality happens if a being does not understand how reality functions in terms of the 3 signs of existence.
Mahasi Sayadaw compares physical matter, materiality, to sea foam (p. 65). From far away sea foam looks like it has a structure to it but when you inspect it up close you will see that it is empty of any form. It is hollow and filled with air bubbles. Modern physics will tell you that solid matter is merely an illusion and that it actually consists of mostly empty space. This is due to the enormous amount of empty space there exists in an atom between the nucleus and the electrons.
If beings were to spend more time in ultimate reality (paramattha sacca) then they might view the situation of an animal suffering differently, as bare experience; "seeing, hearing etc.".
As i said in the beginning i do not have a final answer to this question. Please see only this answer as an input. It might not be as i have described here. If i have misunderstood anything please feel free to correct me.