Re the law on shooting birds, for those 'pest species' under whatever section of the act it is, no licence is necessary as such. A 'general licence' is issued each year by Defra, which simply just 'exists' - you don't need to download it, own it, apply for it or show it to anyone. The licence says that it is legal to shoot eg a Collared Dove where this is necessary to protect crops, livestock, human/animal health, property where no other course of action is available - i.e. a last resort. You must be authorized by the landowner, and use a legal method (eg a legal gun) in a legal way (eg not on a public highway, or in a way that will endanger the public).
This covers species such as woodpigeon, carrion crow, magpie, collared dove, feral pigeon etc. The licence is not implemented in the same way as it is worded, however, in that nobody has been prosecuted for shooting a crow when it clearly wasn't causing damage to anyone or anything - it is assumed that such species are pests, and also that shooting sports require it. So you would be hard pressed to get a result on the grounds that there was no need to shoot the dove. Nobody cares about the dove in law - it is inly really protected from outright cruelty such as torture or illegal methods of taking it (bird lime etc). But firearms offences are much more frowned upon, and this is where prosecutions happen and the police get involved, as there is a public safety issue. So the focus of anyone in this position is not the dove, but public safety and the gun. This is where the complaint should be made - and where the police are more likely to act. Ring them up and say "someone's shot a dove and they weren't authorized" and you'll get little response. Ring them up and say "someone's firing a gun out of their window into my garden where my children play" and you'll get an immediate armed response. They'll want to see evidence, in which case you produce the dove with a hole in it. Let them work it out for themselves, but altimately they have to respiond to your concern about a gun being discharged into your garden.