One way or another, you HAVE to get rid of that beaver. Trees grow back, that's not the problem. A beaver has to have a home. If he cannot build one out of sticks, he will tunnel into the ground. Beaver homes in the ground are the number one reason for dam failure!!!!!!
In California, they have a huge system of levies and canals built up around the Central Valley that we just call the Delta. Both sides are lined with rip rap. basically large chunks of concrete. This is there to stop erosion and beavers from tunneling into the sides of the canals. Every year they have to repair the levies because of beaver homes weakening them to the point they fail. Farms flood, roads are ruined and it costs a lot of money to fix.
California is probably one of the least hunter friendly states out there. Tons or rules and regulations that most people in other states wouldn't believe. But when it comes to the beaver, there are practically no rules. You can shoot them with anything, any time of the day or night, with lights, year round without a limit. I've been out with a State Trapper, and a buddy married to a game warden. It's just amazing how many of them there are out there when it gets dark out and you are shining lights on the water. Eyes everywhere!!!
Back to your pond and your beaver. It is just a matter of time until they dig into your dam. That is the softest dirt for them, and usually steepest. This is ideal for their purposes of creating a tunnel and then a cave big enough to store food and to sleep in. Over time, it will get bigger, and your dam will become weaker. You will never find it, and you will never know it's there until the day it fails. KILL IT NOW!!!!!
My wife and I shoot a couple beavers a year on average. They can be the easiest thing in the world if you find them on land, or at the edge of the water eating. Or they can be extremely challenging if all you can see is the top of their head swimming. I consider myself to be a decent shot, but that's a tough shot. Especially if it's a small beaver and it's over a hundred yards away!!
Do not use a .22 We use our deer rifles. If we have the AR with us, we'll use that, but so far, haven't had the opportunity. We carry the AR mostly for coyotes. Nothing worse then going for a walk and seeing a coyote just standing there watching you at 50 yards. Yes, that has happened.