Thanks to some kind advice from @jkwilborn I unplugged the signal connector from the LPS, to narrow down the cause of my laser not firing.
After I unplugged the signal connector, I powered on the machine. It began homing as usual. But when I briefly pressed the Test button on the LPS, the laser fired on full power continuously until I turned off the machine.
It would seem the LPS and laser tube both work.
Yet when I plug the signal connector back in, restart the machine, and try to Pulse the laser via the controller, nothing happens.
Is that the expected behaviour of the Test button?