Stamina in this series is tricky and is often used more as a hype tool than an actual measurable stat. If Oda wills it, two characters can fight for days on end, only to get downed in under an hour by other characters. Hell, Akainu is likely going to go down to Luffy in that time frame despite him and Kuzan popping off for 2 business weeks against one another. So, imo, stamina can't really be debated properly in the series.
I guess I'll go into more detail about my thoughts on Katakuri's potential performance in this fight while we're still on the topic:
The fight isn't going to be as simple as King just chilling in Flame On mode and fending off Katakuri with basic attacks. From what we've seen, King almost always used his stronger, named techniques when the fire on his back was off. That wasn't always the case, as with Imperial Wings, but all of his physical attacks had him go flame off. And it was during those attacks that Zoro managed to get the drop on King. Even early on in the fight, you can see Zoro's Tiger Trap interception break through more of King's mask, surprising him, likely implying that Zoro broke through his defenses even then. And it's during those attacks where Katakuri's FS will shine because Katakuri can more easily time his counters against King while his flames are off. The large majority of Katakuri's attacks against Luffy were basically counters to Luffy while he was still in the middle of his attacks. Of course, King could stick to the skies and his fire attacks, but I highly doubt that would be enough to take down Katakuri.
King's physical strength is his biggest asset and, while I think Katakuri's Power Mochi would allow him to not be completely outboxed given that that particular technique was still tossing around Boundman during it's short appearance during his and Luffy's fight, it may not be enough as we don't really have a good measure of it's upper limit. I think Mole would be where most of his offense would lie. Unfortunately, because he had to toss it away to give Luffy a fighting chance, we don't have much to go on for it. What little he did show makes me think it'd be enough to deal a significant blow to King when his defenses are down. His haki should definitely be up to snuff to do so. Side note, not confirmed, but as I mentioned earlier, I think the argument could be made that Katakuri may have AdCoC.
Those parallels feel intentional and that scene with Kaido was right before Luffy made his realization that you could infuse CoC in 1010.
Katakuri's Awakening would also be a pretty good boon as it would give him another method of potentially downing King without worrying about bypassing his Lunarian Defenses. Plus it opens him up to a variety of possible avenues to go at King once he's in range.
All that being said, King is not an easy target. I can easily see things quickly going downhill for Kata should King catch him off guard because King not only hits hard as hell, but he can quickly put on pressure with his rapid attacks. And while I think FS is often underplayed way too much, it is not without it's flaws. But the flaw isn't related to speed or stamina. It's simply that FS revolves around the sight of its user. Katakuri got caught off guard by Snakeman not because it was faster, but because it attacked from angles he didn't initially think it would attack from. He got used to it eventually, but that is something King could take advantage of to throw Katakuri off. Especially once he makes use of his suit's hidden weapons. He would be able to keep Katakuri guessing with some attacks.
Earlier I mentioned that King sticking to the skies wouldn't be enough. I stand by that, but that'd still be a clear advantage for King as Katakuri would have no means to reach him (we don't have a clear idea of how far Katakuri's mochi can really stretch) but it would mitigate the effectiveness of Katakuri's awakening. And while I don't think the fire would be enough, King would still be the only one with a way of attacking at a range and holding that advantage for as long as he wants. It wouldn't be enough to take Katakuri down, but it gives him time and options to think of different approaches.
TL;DR: I think Katakuri's moveset is much more suited for King's than King's moveset is for Katakuri. FS especially gives Kata an advantage as it allows him to capitalize on counters, which would allow him to discover King's vulnerability easier than others and his spear gives him the offensive means to deal the necessary damage to potentially take King down.