Fair point about pest control - I'd forgotten they are now a pest species. However if you need to shoot them for pest control (because other means of deterrent have failed of course ), then a sub 12 air rifle is probably not the most efficient way to do it, but as other have said, I can't see anything wrong with a head shot within range.
As to tonnes of shot, I would expect a competent shotgun user to place a pattern of no. 5 shot in the front third of the bird, including the head, as this is considered the best way to fell large fowl in flight, instead of the older method of using heavy l