Moral objections to cruelty
Congratulations for taking a moral stance on cruelty to animals. Even by posting here, you should take satisfaction that you are advancing the moral argument (in Northern Europe/Oz/NZ at least, where the centre of morality is centred - and the rest of the world eventually follows a few years behind).
But as others have said, all you can do at the moment is to refuse to fly the cargo - with the risk of being fired.
But bear in mind that often taking a morale stance, although it may lose you a job, can open doors in other careers.
My first flying "job" was dropping two sacks of Winalot dog-biscuits from a microlight onto the (two day starved) hounds of a fox-hunt in UK. The dogs sensibly all gave up the hunt for the free biscuits. I ended up with a criminal record - but it has been a great asset to me later in my career as it demonstrated my commitment to a worthwhile cause - to defend animals against torment for human pleasure..