How to adjust this BJT design to increase brightness of LED without changing PCB?
All of the current to the LED has to flow through R13 -- this is going to be your biggest limiting factor, particularly because R13 is already at the minimum resistance called out in the TPS2121 datasheet.

I wonder if your former coworker used a circuit designed for a push-pull output, where the TPS2121 has an open drain?

Changing to a MOSFET will only make things worse -- the base-emitter voltage of Q1 will be around 0.6V normally (it's only 0.5V here, probably because of the low current).  A FET would need more than that from gate-source to turn on.

It would be a board mod, but connecting the collector of Q1 to +5V should result in immediate and delightful results.  If you pulled C8 off of the board, you could replace R14 and R15 with zero-ohm resistors (R13 would provide any necessary current limiting to the base of Q1).