I had a war with beavers one summer many years ago. They lost, but it was close.
Don't waste your time breaking, blowing up, or burning their dam: that just encourages them to build a bigger one. Don't bother trying to move the dam to the upstream side of the road: they really want the road flooded and you just can't argue with a rodent.
The only compliant beaver is a dead beaver, a live one will NEVER do what you want.
They aren't too smart about traps if you have the moral strength to use them. I don't mean the "humane" kind.
They don't seem to care about lights, but they are pretty crafty at spotting and avoiding hunters. Thus a night vision system is not needed but you have to stay very still and quiet for them to show themselves. They are smarter than squirrels though and others won't come back out for a long time after you take one.
A heavy barrel "varmint" rifle is ideal as you can stay a hundred yards away from the dam and pick them off accurately. A .22 is really fairly quiet, but is too loud unless you are in a really rural setting and don't have rodent loving neighbours aware of your planned killing spree. Most country folk are familiar with vermin pests and will not interfere with their elimination, but city folk who move to the country to "get close to nature" often have strong objections to killing nature's little darlings.
A crossbow works very well at killing them, is accurate and close to silent, and doesn't cripple your arm or take practice and skill like a longbow. The problem is that beaver dams are always near a large pool of relatively stagnant water (well, that IS the point of the dam) and, consequently, always feature an unbelievable number of small bitey insects - especially after dusk when the beavers start to do their evil work. This problem generally eliminates any qualms about inhumane traps in a night or two - no repellent is perfect and it's truly unpleasant to have to stay absolutely still and quiet while mosquitos buzz about and bite you. You have to be quite close to reliably hit a beaver with a crossbow.
The cadavers aren't that huge a problem. Drop them on a road and run over them with your car... voila: road kill.