How to kill Black Dragons?

Yes, you are right. Sorry, I got fixated on the idea that nothing affects the Champions of the Deep and everything is useless against them except strength and numbers. Turns out that's not true! I just checked myself and was pleased to see how a druid summoning mantises can tear these strong guys apart.
So, it's mechanical damage. And it also reduces the movement speed; the shackles also reduce the Champions of the Deep's damage by half. But! Any action other than an attack will free the bound target. Therefore, if you've bound them, finish them off, as they say.