From Fallen London Wiki

The contradiction embedded in that Quality is that people will *call* you the Fist of the Bazaar when they really mean you're the Fist of the Masters.

In fact, at least in the Affair of the Box (which currently grants the Quality), both Stones and Fires are working cross-purposes to their employer. The former wants out, even if it means rushing the Bazaar's project. The latter wants to preserve London, even if it means ruining the project entirely.

Long story made short? Choosing to side with the Bazaar *against* the Masters is, ironically, not a very Fist of the Bazaar thing to do.