GENEVA—Blinds were drawn at a back office of the Intercontinental Hotel one day last year, as a man known to staff as The General emptied a white cloth bag stuffed with euros. It was time to pay for one of the world’s longest-serving leaders.

For weeks, several former hotel employees recalled, the five-star hotel had been locked in a secret routine serving its best two customers, the first couple of an impoverished oil-exporting Central African country. Paul Biya, president of Cameroon since 1982, and his wife, Chantal, clock...