That is, when you're actually playing the hero type. Just why would one go from saving the world from a Daedric Prince to running errands for another?

A neutral or evil character might just not give a damn, but a morally good (not perfect, but still certainly in that category) probably would more than hesitate when faced with the idea.

It's even more complicated with a character that has become a divine crusader. S/he would steer clear from that door.

Did you mean SI, as in, Shivering Isles?

Anyway you could role-play it as something like; after all of his ordeals in saving the world and doing good, he's just become tired, weary and slowly corrupted. Since it is generally accepted as canon that the Champion of Cyrodiil becomes Sheogorath, might as well play along... just my 2 cents.

Yeah, it may take some work and time before your character becomes someone who would be more willing to take up the offer, but it's definitely manageable.

But KoTN and SI are fully incompatible roleplay wise, as it isn't only a matter of morality or contradiction, but also the simple fact that the divine crusader is a servant of the Aedra. It only makes sense if you leave their ranks after becoming unworthy of wearing/using the relics.