Wouldnt that be more to do with the appearance of the sangha to the wider society?
No. It's an offence of defeat, because it fulfils the conditions necessary to be the unwholesome kamma of urging another to kill a human being. If, with compassion, the bhikkhu had said, “Please don't kill him,” but the executioner went ahead and killed him anyway, the bhikkhu would not be guilty of any offence at all.
I agree in that case, but thats not relevant to mine where the bird would have died anyway but more painfully and slowly.
The motivation to heal it would could also be rooted in aversion to the birds suffering
Not at all, it would be rooted in love (adosa) and compassion.
Like my motivation to kill it...
The outcome is always the same. One scenario has less dukkha and the other scenario has more, but the one with less dukkha is morally wrong and the one with more dukkha is right?
The root of your dilemma may be that you don't believe in kamma and rebirth. Looked at with the wrong view of "one life only", suffering ceases at death, but that is not the Buddha's way to end suffering.
I have never said that there is one life, therefore the rest of your argument is invalid because its based upon a false assumption.
A seriously injured bird will die of its injuries in due course. The best that you can do is remove the immediate danger of the cat tormenting it further before killing it, and keep the bird in a safe place. Those are all compassionate acts. Breaking its neck is not a compassionate act — it is breaking the first precept.
Its an extension of compassion if the death was inevitable. Why should the bird suffer a long death?
As I said, the outcome would have been the same. One would have brought more suffering, the other less.
Why should the bird endure unnecessary
suffering before moving on (to oblivion or the Deva realm)?