As long the paladin stays close to the king he can assume a position of power to use his influence to advance good and to prevent the king from committing crimes. Maybe his his commitment to the late monarch obligates him to lead the new king back on the path good. If, on the other hand, he cowardly flees because he's afraid of succumbing to the ill influence of the ruler, or afraid of losing his divine blessing, he does not act as a champion of goodness and justice, or the deities representing these principles.
It is the duty of the king to justly rule the realm and it is the duty of the paladin to defend order. A king who fails to uphold justice and to protect his subjects is not fit for rulership and a paladin who fails to defend kingship and kingdom (possibly even by removing an evil holder of the throne) from corruption and evil fails as champion of justice. The life of a paladin is a series of divine quests to prove his quality. If he lacks the courage to risk his position or even his life for good he is unfit to be a paladin.