The word 'cull' is used in the natural sciences, and is a more appropriate word to use in this case, as it carries with it connotations of removing an animal from a gene pool based on undesirable genetic traits or defects. Like euthanasia, culling almost always implies putting the animal down as a means of doing so. It doesn't add ambiguity at all, in fact, it gives us additional information.
Cheers,
-Matt