Theophilius Bagbiter is an independent contractor associated with the Criminal Faction. His card art depicts Bagbiter at a table showing holograms of a playing card and of the secretive runner Reina Roja. It's possible that he is a gambler and/or an information broker of some kind.
Bagbiter clearly takes inspiration from the Mad Hatter as demonstrated by his top hat and his quotation of the unsolved riddle from Alice in Wonderland. Theophilus Carter was the name of an Oxford hatter thought by many to be the inspiration for the Mad Hatter.
"Why is a raven like a typing desk?"