A privately owned Typhoon still answers to a flag state

What rules would govern capturing or sinking a privately owned Typhoon-class submarine?

Treat it as a registered private vessel unless it supports unlawful hostile activity; private ownership does not erase flag-state jurisdiction.

Private ownership alone provides no reason to capture or sink the submarine. The example of grounds for action is assistance to a foreign power engaged in illegal activity or under censure. If action were justified, the military problem would resemble dealing with another submarine, although a Typhoon's hull structure would make capture difficult. A private owner must still register the vessel under a flag and follow the relevant state's laws. Private ownership does not create a sovereign-citizen exception at sea.