Kaidou didn't say he was afraid to start a war with Big Mom; he said he wasn't prepared to fight her at the top of the waterfall, which is different. We saw that part of his resources were gone with Apoo, who returned only three days later.
However, when King offered to go, Kaidou simply changed his personality and became calmer because a combatant who had the ability to fight them at the entrance was present. The problem was never fighting Big Mom because we saw Kaidou freeing her from her imprisonment and challenging her to a fight to the death, which shows Kaidou's high confidence in defeating her.
Kaidou was also going to an enemy territory filled with enemies (Marines and Whitebeard pirates) with only King by his side, and it is explicitly stated in the manga that Kaidou, with only King by his side, engaged in conflict with all the Red Hair Pirates, including Shanks, in a 2 vs. all battle.
After identifying the strength that Big Mom was bringing with her, King made fun of her for bringing only one ship with her, saying, "You expect to face us with just one ship?"