Simple.
Have the animal or creature be the projectile.
Flight is the best way to carry a projectile over the walls of any location. Even the strongest animals can only throw an item so far. Birds, however, would be able to easily fly over the walls of most encampments. Then they could easily dive down into the city, lock on a target, and strike anyone that moves.
Peregrine falcons are able to dive at remarkable speeds. I believe some have been recorded falling at about 240 miles per hour. Imagine a massive bird of prey or an even larger animal flying down and falling straight on top of the encampment of your enemies. All the creature would need to do is fly at a high enough altitude, then stop and let gravity do all the work for it. It would effectively be a living missile.
The people making these creatures could come up with all sorts of horrible additions to the creatures as well. Spikes, spines, or other protrusions on the body would make it such that, once they fall on top of the person, they effectively skewer them. You could go the opposite way too, and simply make these flying dive-bombers as massive as possible to crush the opponent.
A particularly devious idea would be to have these creatures be particularly fragile. When they hit the ground, they burst open, spilling a vile substance that was locked inside their guts. What is the substance? It could be any number of things. Acid comes to mind. Human gastric acid is pretty dangerous on its own, so imagine a creature where its guts are designed to be as acidic as possible. Anything that touched its stomach acid would not be looking pretty afterward.
Poison is another option, but the most devious idea by far would be hiding a virulent pathogen inside these creatures. When they die, the creatures release it into the air and the specially designed disease would sweep over the city. You'd bring your enemies to their knees pretty quickly.
Presuming the attackers have the only cure, the city would have no other option but to surrender if that was the case.
With ranged attacks, flight is by far the best option, and you can get extremely creative with it. Give the birds rocks to carry, and they can dump them on the heads of their opponents. Forget catapults and cannons. Imagine a flock of birds raining down a thousand rocks on your opponents. Going back to my acid idea, imagine thousands of birds that spit toxic acid. Then, imagine that raining down on your enemies.
If birds are not satisfactory, insects are more than capable of filling the job. Not only are they horrifying to behold, but they can swarm en masse and attack everything they see. Once they're done wreaking havoc, go back to the pathogen idea and have them die off, carrying some horrible disease to every corner of the city. Anyone bitten gets it. Killing them is not enough to stop it, though, because just being in the presence of their corpse spreads the ailment.
I think such an attack is far more practical and scary than hurling a massive rock at your opponent, but there are ways to do that as well. You just need something big enough to pick up a massive boulder and then fling it in whatever direction you point at. It just needs a grip. Giant hands would probably be fine.
You might also have a giant lizard with a tail capable of holding a rock. That would be intriguing. The tail would flip back and then release the stone as fast as it could. Tentacles in the shape of trebuchets would also be interesting, though I'm not sure if a giant squid would be able to hold such a thing.
If you're looking for sheer effectiveness in combat, though, my recommendation is giant birds that spit stomach acid on their enemies below. Then, when they are unable to attack anymore, they go as high into the sky as they can and dive onto their enemies with as much force as possible. When they die, their toxic insides sicken their opponents and carry virulent pathogens that only the people who made them know the cure to.
That's the best I can come up with. I hope it was a satisfactory answer.
