If this 'permission to build' was being done by a normal US administration, I'd wonder if it was a chip on the chessboard--letting korea, in the face of dprk and maybe china, move closer to nuclear weapons without actually having them (so no issues with the NPT).
It may have a similar ring as what has happened in japan. Takaichi welcoming the guy from taiwan, and kind of at the same time saying problems in taiwan would probably merit some reaction from japan's self defense forces.
If it did act to defend/help taiwan, I'm sure the US would like to quickly use (or have available) any/all of its military assets in korea to do that, probably pulling korea into it. And if korea could field subs, even if not directly involved, it'd be more noise/targets out there for the chinese to deal with. And the thinking might go that if there are more subs out there, with the chinese likely unable to discriminate which are US, or korean, or japanese, they might be more cautious about taking any action.
blah, blah, blah...