Well, this was resolved by getting a cat water fountain (actually two, because the first one stopped working well after a few weeks).
Now the most annoying thing he does (other than jumping onto the counter when I’m cooking) is attack my hand while I’m using my mouse. When I’m downstairs, he’s satisfied to sit on my lap or play catch, but when I’m upstairs, he inevitably jumps up to my desk, sits on the mouse pad and bats at or bites at my hand while I’m trying to do something on the computer. I can play catch with him for a while to keep him off, but eventually, he loses interest and goes back to the mousepad.