The comment about an inrō on page 9 is a reference to the long-running jidaigeki (period drama) TV series about Mito Kōmon (水戸黄門), which ran for 42 years. In the series, the titular Mito Komōn carries an inrō with the seal of the Shōgun, which he displays at the climax of each investigation, to validate his authority to place the culprits under arrest and otherwise deal justice.
In other words, Erik is being asked for some kind of token to show that Luck is acting on royal authority.