First off, here's the thing: FS can be countered by superior speed, who can tell me for sure that King with his flames off can't outpace Katakuri's FS? Dude, Kaido was doing exactly that the whole time against Luffy, he was wrecking Luffy who was using FS non stop, without even using his own FS on the rooftop, with nothing but pure attack speed, movement and reflexes. He was only forced to use FS once because Luffy caught him off guard while he was having a flashback and lost focus, which makes it debatable what would've happened if he hadn't been caught by surprise.
It's also pretty wild to just assume King has to keep his flames on the whole time and take hits nonstop until Katakuri runs out of steam or his FS weakens. Who can prove that King with flames off can't reach the same result? Even without the flames we're talking about a Lunarian plus an ancient Zoan combo, that's more than enough to tank everything Katakuri's got. He can just straight up sacrifice his invincibility and still be able to withstand just as well, no problem, which makes the fight really tough for Katakuri, because with flames on Katakuri can't deal damage, and even with flames off King is still able to withstand and tank. And if his speed in off mode, both attack and movement, is able to outpace FS, then it's gg.