I would rather catch them in the winter when the fur is prime and they don't get chewed on by turtles. But they turned up in the farm pond and need to go...
I had a hell of a time getting that pipe unclogged. After I got the flow going, they when up in the pipe 7-8" and sealed it up. At first I cut a pole from a dead treelimb and thought I can jam it a few times and break it up. Didn't work. So I got a fruit picker on a 10' pole, and tried to corkscrew it with the basket. I was up to my shoulder and had only managed to push it in further. Finally I got a 20' telescopic pruning handsaw, removed the blade, and duct taped a potato rake to it. By this point the blockage was about 15' into the pipe. I was able to rotate the rake against it and piece by piece grab off twigs and debris. Finally I got a got a good hold on a chunk of debris and that got the flow started again. Kept at it as much as I could and opened it up good. There is still some junk in there about 20' in. The PVC pipe goes down through the dam. So it I couldn't get it lose it would be an expensive fix.
I don't know if there's anymore beavers in there, I guess I'll find out in the next few days.
Easiest thing to do in perpetually clogged pipes is to leave a piece of chain or cable in the pipe. When it gets clogged, "pull" a small tire through it and attach another chain/cable to that end to leave.