I think you have a duty ( and I say that with no intended lack of feeling or flippancy either ) to slot them at every opportunity that presents itself. They do not belong here, they are an invasive species that does a huge amount of damage to native fauna and flora. I also hear there very tasty. If I lived closer I’d gladly dust off the old faithful Webley vermin wollop’er and lend you a hand. Helping the environment, practicing some skills and a free feed what’s not to like there really.
But just as that wouldn't have been good for Hadely's Hope, it wouldn't help Broch much with preserving his trees and birdies!Nuke 'em from orbit - it's the only way to be sure.
I don't like to see introduced animal species (without their native checks and balances) introduced into any established ecosystem. I read that your native squirrel is being replaced by this invasive nuisance. .22cal LR and harvest them in a sustained yield fashion for my table. No shotguns. Too noisy and pellets (lead?) flung all over and embedded in the trees. I'd take a little pride in marksmanship.
Plus, I've been trained for so very long to always consider the background to any shot. .22cal shorts are hard to find here and don't load properly. Each cold shot has to be a counter.
I would think a nice savory curry with a plum sauce could be a treat. We can argue about the wines at the table.