It's worth noting that when katakuri lost his cool and his observation haki failed him he took multiple hits from boundman luffy. Of course, once katakuri regained his composure he was able to deal with G4 at that point without much trouble. What followed is a bit wonky because luffy had ample room to test and develop his observation haki to the point that even without G4 he was managing to dodge some attacks and even land some hits, which is a clearly dramatic improvement from before. As in, with boundman which is dramatically faster than his base form he was eating hits from katakuri early in the fight while in turn even with his accumulated damage he managed to evade attacks later in his base form. At the end of the fight luffy's own observation haki was quite close to kata's.
Stamina wise it's worth noting that zoro got that mink medicine before his fight. Whereas king went straight from fighting marco alongside queen to his confrontation with zoro. And by the looks of it marco did manage to land some hits (even even through king's defense mode apparently, to marco's credit).
Another thing worth noting is that towards the end of the fight with zoro king was dodging attacks even while in defense mode. Can katakuri do damage through king's defense mode? There's no real limit to is as far as we know.
Yes for Kata's defense to work he needs aCoO which means he has a few weaknesses to his defense in comparison to King's defense. For one Katakuri needs to have haki so fatigue from aCoO overuse can allow him to get hit and he also needs to keep his mental/emotional composure which can also be an issue though with King I don't know if that would happen. Outside of that the opponent would need to have aCoO themselves to counter Kata's defense or have that much better speed. Luffy's growth in that fight is not likely something King could easily duplicate so I don't think that would work against Kata. Additionally, Kata's defense with aCoO does not cost him his offense whereas King can't attack at full strength and keep his defense up either. If King tries to attack Kata would definitely use aCoO to counter and do damage.
Stamina wise I do think they should be roughly the same. The only fights we saw King and Kata in lasted essentially 10-12 hours. Zoro did have the advantage of the super drug but he also had a broken body and a time limit on the drug so I think everything evened out. Luffy in the Kata fight had unfair advantage of getting to use Brulee to escape and rest to get haki back and he also happened to eat due to Kata having a food based DF. Both of those are things King would not be able to exploit.
I think Oda has not been consistent with defense mode. With the Seraphim it seemed pretty much absolute. But could also be that in the end of the fight King was using attack mode to finish Zoro off so he had to dodge or alternatively aCoC can bypass the defense mode (we just have to believe both Luffy & Zoro conveniently forgot against the Seraphim).
My point in the Kata v King fight is that King can't land a hit on Kata easily without either wearing him out (stamina which I think both are about the same in with King having slight advantage due to zoan) or making him lose his cool. This would leave King with only one option which is use attack mode speed to try to land a hit on Kata since high enough speed can overcome the aCoO. However, that would leave King vulnerable to counter attacks which Kata specializes in. So if King is not willing to risk it then he stays in defense mode which Kata can't bypass. This would leave them both at an impasse where neither can hurt the other and just have to see who tires faster. I just don't see King fighting in this way which is why I give Kata the overall edge in the fight even though it would be close.
Other things to consider is Kata being an awakened user has alternative win methods to win too. He could in theory bury King under a ton of mochi and crush him, suffocate him, or at the very least immobilize him. I don't think King has the same options against Kata since nothing he can do is not in theory dodgeable.