A bilge pump is exactly what it says. I can think of only two places a bilge pump could discharge. Overboard or into another bilge. I can't think of any reason why they would discharge into the firemans passage.
Could this "other mischief" be the result of an egregious mistake the crew might have made in activating the wrong valve of the ship's bilge pump network in which water was pumped from a flooded compartment only to be released in the Fireman's Passage?
During the limitation of liability hearings, Harland & Wolff ship designer Edward Wilding would comment as to why there were two doors located in this area:
We have a very strong prejudice at Harland & Wolff's to having a watertight door actually adjacent to coal, which has to be worked during the voyage. No. 3 hold was used as a. reserve bunker. In order to maintain the intactness of this D bulkhead, which is at the after end and yet enable coal to be got out of it at sea a watertight box was built over the after end of the pipe tunnel, having ordinary non-watertight doors opening into the reserve bunkers and having "a watertight door on the after side opening into the stoke hole. So that, if necessary, reserve coal, when being worked out with the reserve bunker nearly full, did not come in contact directly with the watertight door.
Just inboard of the shell in cargo hold no. 2 was a longitudinal bulkhead to the firemen's passage. This bulkhead was jogged outboard for the access trunk from the crew space above. When the collision came the outer strakes of the transverse bulkhead were thrust inboard. The riveted connections between the longitudinal bulkhead and the main transverse bulkhead were compromised so that rivets failed and a seam was pried open, allowing water to flow in. This is what stoker Charles Hendrickson saw when he was in this access trunk after the collision."