In the Carolinas if you have 1 beaver you have a dozen, and once you kill that dozen wait 6 months and you will have a new batch move in.
I did that as a side gig while living in North Carolina. Golf course and gated communities have a very low tolerance of giant rodents that eat expensive ornamental trees.
As to turning them into food / those smaller than 30 pounds eat the best. Once over 40 they can get tough and musky, and often are very scared up. They are pretty much aquatic alley cats that build dams when they are not feeding fighting or fornicating
As has already been mentioned / skin but stay away from the castor glands and oil sac. Once that touches meat it’s wasted.
If I wanted to eat one - I would make my incision down the back and pull out shoulders, saddle and part of hams.
Favorite way to prepare is sous vide then quick pan sear followed by coating with mushroom/bacon paste and wrapping in pastry to make Beaver Wellington
Without seeing the water they are inhabiting......if you can make a breach in the spillway or overflow, it doesn't have to be big, but bigger is faster, they will come running to their death to fix it. Just be there when they show up.
Also, if you have access to a full auto M1 carbine, it is a beaver killing machine....or so I'm told.
I knew, at some point, the tread would get crashed by the 3rd grade euphemisms. By the way, be careful if you use voice-to-text to message a friend that you have a "beaver depredation permit", Google will choose "deprivation" every time...