Very sad about the OP's cat. If they've actually recovered a pellet then that's pretty solid evidence, but as they say, pretty much impossible to pin down on anyone without witnesses. Actual airgun shootings are pretty rare. My brother is a vet and he gets any number of cats in that have reportedly been shot. On x-raying, the vast majority (8-9/10) have just got a nasty puncture wound from fighting or some other misadventure. Unfortunately "air gun" is the go-to explanation for people who don't want the expense of x-raying and as such is strongly over-reported (especially if the animal has died). It does happen, but thankfully not as much as people think, or there would be a lot more injured cats.
The mis-diagnoses occur because actually, unless there's internal bleeding or something that needs dealing with, trying to extract a pellet will do more damage then it's worth, so they leave it alone, and don't realise actually there's no pellet anyway, they just sew up the wound and pack them off. Surgeons in major trauma units frequently leave bullets in if it's going to be unneccesarily risky to get out. Same principle - they've done all the harm they're going to!
To deal with a few of the misquotes on airgun law:
"If it was a pellet it was an air gun rather than an air rifle and the laws are very flakely for them. Vets regulary pull pellets out of all kinds of animals, but particularly cats unfortunately."
The what hey?
Air gun is a generic term for an air rifle or air pistol. In technical gunsmithing terms a "gun" is smoothbore, as opposed to a rifle which has a rifled barrel, but there is no such distinction in law regarding airguns - only between rifled cartridge firearms and shotguns.
I think you're mixing up air guns (which shoot pellets, and are deemed firearms, but not required to be licensed), and airsoft type ball-bearing guns which are generally low-powered enough to be classed as toys (although subject to the VCRA if they're realistic) . The distinction is that you can commit firearm offences such as armed trespass with an airgun. You can't with Realistic Imitations (although there are some separate offences for RIFs under the VCRA).
Thixotropic - an air rifle does not require licensing if it's muzzle energy is less than 12ft-lb. 15ft-lb will put you in chokey! For air pistols it's 6ft-lb, and you can't have an overpowered one, because that would be classed as a pistol, which is banned. Almost all air rifles are under 12ft-lb. The only ones over that are specifically tuned up for people with a need - such as pest control in or around a stables or built up area that would normally involve a cartridge rifle, but where in the case at hand, that would be dangerous or too noisy. 12ft-lb is more than enough to take rats, rabbits and small game, so licensed air rifles are pretty rare (and expensive) for very specific use-cases.
The rule on use within 50m of a highway is that it's an offence if it disturbs or alarms road users. That means if you're sat behind a hedge backing onto the road, you're probably fine as far as cars go. It's only a concern if there's a pavement on the other side of the hedge that people might be walking down. You also have to be wary of alarming horses. It's a bit of a stupid rule because it's better to be next to a hedge bordering a road shooting INTO a field than on the far side shooting towards the road! Of course it all depends on the layout of the field - hills, banks, whether you've got a high-seat that allows you to shoot down towards the ground (the planet makes a pretty good backstop).
There is a consultation in Scotland on licensing airguns (they've backed off on a ban after they realised that something like 30% of Scotland's Commonwealth Games medals come from the Scottish shooting team). Simply put it won't work. Unlike the pistol ban in 1997, the Police have no list of who has what. In 1997 if you didn't hand you gun in by the deadline you got a tap on the door. With a sledgehammer.
If they introduce licensing, then if someone chooses not to license their air gun and also chooses not to hand it in during the pre-licensing amnesty, the Police literally will not know unless they do something naughty with it. How could they, unless they do a house to house search on every property in Scotland!? The licensing system will rely entirely on people being honest enough to license their airguns or hand them in. Which means the honest people who aren't a problem will do exactly that, and the dishonest people who are a problem won't. And those guns will swill around in a Scottish black market (also, how do you stop people buying airguns in unlicensed England and taking them north!?!? It's why Alex Salmond's dream of doing his own thing regarding firearms legislation is a media-showboating waste of money).
In England and Wales the problem would be even more pronounced with an estimated 10million air rifles and pistols swilling around the country (one for every other home!). And the Police have absolutely NO idea who owns any of them (bar the few that are already licensed as overpowered).