Ok!
I've worked out enough of the pinout to get the laptop running with a third party power supply. I don't have a good battery, so I wasn't able to really determine what the battery sense terminals do, or see what things look like when charging.
In the image you can see the pinout looking into the computer. These are the male pins.
1) Battery Sense - I don't really know what this signal does, other than it is related to the battery
2) Battery Sense - I don't really know what this signal does, other than it is related to the battery
3) PS_ON - Connect this signal to ground to turn on the power supply. The laptop doesn't care about this signal, only the power supply.
4) +18V - This is the main power feeding the laptop. This is at about +6v when PS_ON is disconnected.
5) Voltage Sense? - I believe this is to compensate for voltage drop across the wiring. When I disconnect this with the laptop powered, the voltage produced by the power supply increases slightly.
6) GND
I confirmed the computer will power with just 18V connected to pin 4, and 0V connected to pin 6, and all other pins open. This was tested with an SLT/386 and a bench supply.