Corporate jet Captain reported receiving a low altitude alert from ATC on a night visual approach to BWI; citing a possible false glide slope as contributing.

Narrative

We were an hour late on our schedule after a late passenger and deicing in BJC so had to quick turn at ZZZ to repo to TEB and get the aircraft serviced before a late flight with pax back to ZZZ. Descending out of 15000 into TEB we got a divert message to BWI. We managed to get the divert accomplished but when cleared for a visual for 33R they asked us to turn inside of 5 miles so I began my base turn at 4 miles while configuring; hand flying and trying to reference the runway visually (the PAPIs were dim and indistinguishable) so I referenced the displayed glide path and began my descent out of 2000 (FAF altitude) and lost the runway as I was cross checking the PFD and map. Tower gave us a low altitude alert and I immediately added power (flaps2 and gear down) and began a climb out of 1400 feet. I was able to reference the runway again and we were low and I suspect I had a false glide path and was too far to the right of center line to get accurate indications regardless. The rest of the approach was normal.I am realizing we go from delay to massive schedule compression without taking time to take a breath; sit down and eat; and accept diversions at transition to night without considering the risks. I intend to be more disciplined in the future and pad these events with the time I need; either asking for a full approach at night or calling in to delay a departure.