Yup, that's the one. The fifth avocation; i.e. king of the world, refers to the plans of the fellow the narrator is researcher however. And it's all a fraud anyway!
It seems the Man Who Would Be King is an avocational sociologist too, just like the narrator. He has four jobs already; if you do well enough in a certain avocational field you can be promoted into it's vocational branch. He forged and planted all sorts of documents regarding his 'paternity' and 'lineage', launched the king schtick, created the social movement involved, and eventually plans to reveal his 'experiment' to claim his fifth vocation in sociology.
Sadly, the AAmen(?); the AI robots that really run things, have different ideas. They decide creating a human constitutional monarch, along with all the flummery that entails, is just the thing to give the masses something to cheer about.
In the end, the plotter gets his fifth vocation. It just isn't the one he was aiming for!
Have fun,
Bill
P.S. While the plot was nice, I liked the idea behind vocations and avocations, with a mandatory time spent in each weekly, far more interesting.