Even Zoro never did damage to King with his flames up
The difference between the pacifista and King is that while the pacifista seem to always keep their flames up, King lowers his flames all the time.
There is a trade off in regards to the flames. Flames up= near invincibility, but they aren't as fast and their attacks seemingly aren't as strong. All of King's strongest moves required his flames to be down, like his beak missile or the giant flame dragon.
The benefit with the pacifista is that their AP doesn't come from their physical body, like King's does, it comes from the lasers vegapunk has installed in them. As a result, the pacifistas probably never need to take their flames down, while King needed to take his flames down to fight Zoro.