The wildlife and countryside act 1981 has a list of ways snd methods of taking birds and mammals that are prohibited. This include bows and crossbows when used with arrows.
There is no mention of catapults using stones, or round pellets. Indeed you could argue that the device in the OP is some form of crossbow or speargun, but it is not using an arrow but a metal ball as a projectile.
In the the above act there are also clauses that you must not cause unnecessary suffering to an animal or bird.
There is also an Offensive Weapons Act and I think also in the recent Crime and Punishment legislation there were a number of devices included - types of knives, knuckle dusters etc that are now prohibited. Whether the OP’s device is included in this list I have no clue but would be worth checking.
Much of UK law is written as prohibitions, rather permissions.
It’s written that if you use a bow and arrow to shoot an animal you cause an offence.
However there is no mention of a catapult being prohibited therefore you can use it.
But if you inflict unnecessary harm then you commit an offence. It’s up to the user to ensure that the device brings about a humane kill.
