How did you deal with... "that" in your roleplay?

Sure, a job's a job and all that, but I wouldn't think it's quite that simple assuming that your character had always been loyal to the Empire instead of having a sudden outburt of imperial love on a random day and joining the Legion.

Perhaps your character thought Mede was no longer fit to rule, he's a fake, a phony, or perhaps that the Empire would benefit from death.

He firmly believes that the Empire is needed to stand against the Thalmor in the future, but not under Mede. Perhaps under the Dragonborn himself, or someone else, but Mede was a tired old man that was no longer fit to be a war leader.

Loyal to the idea, not to the man.

Short and simple.

Well, certainly not all of my characters have been supporters of the Empire, and my assassin didn't really have a view on the war.

Night Mother and Sithis > Ideology.

An old balding man is hardly the empire