HERE IN THE LAND OF THE FREE..For what it's worth I do know people who have gotten permits to remove beavers after contacting DEC. I don't know the details on what they did to get it. I believe there was a town bridge nearby and it was threatening that. Again, I don't know the circumstances.
Seeking out trappers would also be a good course of action. I was actually thinking about trying to get my trapping license this year.
I'd get a good Air Pellet Gun (quiet).
Sight in in real good, then visit the Beavers once in a while.
And not tell anyone - especially the missus (big mouth).
You should never tell the Missus when you are out hunting beavers. it just doesnt go well.