years ago we had a licence to shoot them on a trout farm , early morning plotting up in the centre of the holdings and we soon bagged them up but you have a large water to contend with which makes it awkward.
like you said you never see them on the bank , does the water have any islands you can get out to? if not it may be worth putting a raft out within shooting distance and maybe making it so it has a backstop on it.
its a big ask but you could pre-arrange to have the lake closed for two hours from dawn but i doubt they would go for that
oh and they stink :sick: