I wonder if the 3I should copy ancient China and have a Censorate. I just read
The Traditional Chinese State in Ming Times (1368-1644) and on Page 50 of that book they describe the function of the Censorate. It seems to me that a similar office, also focused on curbing corruption would both give the setting a distinctive feel and be an interesting Patron for PCs.
The Censorate might send them anywhere in the Imperial realms to hunt down corruption. The lists of foes and distractions would be endless.