Using “it” for an animal is perfectly reasonable. An argument can be made for using they instead but we even use it for humans occasionally (e.g. Jack checked on the baby, it was sleeping peacefully).
Its more “like first weigh the possible harm versus the potential benefit and do not pursue treatment avenues that are more likely to harm than do good”. If it were truly just “first do no harm”, medicine would not be practicable, because any intervention carries risks of some nature, don’t even get me started on oncology.
It is also worth noting that “first do no harm” is not even all that close to the wording of the actual oath, originally it is in fact more like what I typed out above.