Not even remotely. It’s been said by multiple different characters within the BMP that if Big Mom dies, Katakuri becomes the captain. Not once have he and Smoothie been suggested to have mutual respect/hatred towards each other, and Katakuri has been definitively stated to be stronger than Smoothie.
King > Queen has never been stated by any character. King has never been stated to be the second strongest of the Beasts, nor has King been said to be the definitive captain of the crew should Kaido fall. Who’s Who showed King the same disrespect that he showed Jack.
King has never been said to be any stronger than any other Calamity, while Katakuri was definitively stated to be the strongest Sweet Commander.