I've killed multiple beavers and muskrats using a .223 and Nosler 55BTs, buuuut I was preventing pond damage and not concerned with pelts.
You might want to look at using a shotgun also, 6 shot does a number on both rats and beavers as long as the beavers are close in, 4 shot is even better. Most effective time to shoot with a rifle them was dawn & dusk, shotgun with a light at night works great. A high velocity varmint bullet pretty much fragments when it hits water up close, but I still always took shots with the animal against a bank. This is why I'd rather use a HV varmint round over any .22 rimfire around water.
Honestly the most effective way to get them is with traps. When we initially bought our place complete with 9 acre pond and beaver infestation I went the rifle-shotgun-back-hoe route at 1st (stopped short of explosives, but that's how frustrating the little %^^$# are). Then I went the Jeremiah Johnson route and actually had much greater success. Traps while not "fun" are a 24/7 weapon.