Well, I'll wait until the raws come out but I see three ways possible -
1) you need to attack King when his flames are off. [finding the right opportunity when King uses his speed form]
2) you need to manually affect that flame. [Marco's flames or foxfire style]
3) you can hurt King in that form but he'll still be able to switch back into his most durable form soon.
Which leads to two scenarios -
1) Zoro attacks King when his flames are off. (either due to finding the right timing or by switching it off himself with Foxfire style)
2) Zoro with his insane AP now thanks to Conquerors coating overpowers the durability even with the flames on.
For the same:
1) Even when King's flames are off, he's still immensely durable thanks to the Ancient Zoan devil fruit - something Zoro wasn't able to damage [
1]. Same Zoro was casually slicing up Kaido on the rooftop, with stronger attacks ofcourse. Katakuri does not have this level of AP, outside of Mogura that re FS base Luffy dodged.
2) Then the case is just self explanatory.