Kathy: You are probably wise to just avoid that job. After you've caught 200, 300, maybe 400 beaver, you might be able to catch the smarts ones. For example, putting sticks in and around your bodygrip traps is in fact a good way to funnel "ordinary" beaver into the trap. You do that around an educated beaver, and you might as well hang up red flags all over the place.
When I have a really smart one, I employ the same tactics that Mike described. Most of the time, this involves a foothold trap because it's "invisible" and that's what it takes for smart ones. I get in the water way away from the site where I want to make the set, carrying all my equipment with me. A blind set is super. The trick is to not leave any disturbance on the bank.
As far as shooting them. It certainly is expedient. However you have to be a good enough shot to pull it off.
Endeavor to persevere.