So I've got character X assigned Ophidian and a Construct from a Power Battery with the White Ring attached. Character X is dealt enough damage to KO it. It has not used the once per game granted by the White Ring yet.

What happens?

You would "turn to its last non-KO click and it can use Regeneration during your next turn." Just like the wording says. Essentially you would not reach the KO click, therefore you would not remove figure x from the game.

But how do I know Figure X would be KOed without turning it to the first KO click? If do that, isn't Ophidian removed from the game?

You know because you actual do click it to a KO click. However as "would be KO'd" happens before the KO the dial is turned back to a non-KO click before the KO triggers. And thereby the character is never KO'd.

In essence, you would turn it to the KO click, see it is KO'd and instead declare the power from the white ring. Then, turn it to it's last non-KO click and carry on the game.

The power stops the figure from dying. If the figure was KO'd and wiped from the game before the power activated, it'd be pretty useless now wouldn't it? I think you're just being a bit too precise with the KO ruling you linked

In my personal opinion, Ophidian wouldn't be removed from the game because the character it's attached to was not KO'd and Ophidian itself did not reveal a KO click.

Does "Possessed" care about an actual KO though?

Also, since in all likelihood, this occurred during my opponent's turn, couldn't s/he decide the order of effects triggered by my character turning to a KO click? Ie they could decide "ophidian is KOed" resolved before "do you want to use the WL Ron?"

I agree, I'm just trying to make sure I understand how and why this works the way it does.

Now this is an interesting point, however I think, say in a War of the Light event, I'd rule that the Ophidian doesn't get KO'd because the figure doesn't. In theory, the power ring stops it from ever being KO'd, so the entity should still be attached to it.

I would think this applies the same concept as the bizarro + entity. When the click stops on spot 12 then the entity is KO'ed but if it passes by nothing happens.

Your dial never lands on a KO slot. With a When ... would be KO'd power triggers what happens is you turn the dial and see the KO slot, but you don't actually land on that slot. Once you see that it would land on KO you instead activate the power.

Since your dial never stops turning on a KO slot you do not lose the Possessor.