Just check which pin on the motherboard is connected to ground (check continuity with any other ground on the board, don't measure voltage), that's the (-).

The pin on the right (#4) is ground (above the L from EXTERNAL).
The pin next to it ( #3) has +5V, don't connect to it.
The pin on the left (#1) is your +.

1: +4.5V (in)
2:
3: +5V (out) - don't connect
4: GND

Almost always the red wire is positive and the black is negative (I have seen this reversed, but it's VERY rare and I wouldn't
expect it from Compaq.

But you should take the pack apart (should be easy to cut/remove the black film holding it together), for two reasons:

- See what kind of cells are in it - if they are NiCd or Nmih that means they are rechargeable - putting dry cells might not be
good as they typically don't like the charging current. Fortunately it would also mean they collectively produce only 3.6v and
you can most likely add a diode in the line to prevent "reverse-change" (and drop the dry cells to a much closer voltage)

- Confirm the wiring of the cable - typically the Negative end of a cell is simple and flat.. the positive end
usually has a bump or an area inset in from the edges to isolate it from the metal cell casing.

Confirmed this works with 3x rechargeable AAA batteries with + on the #1 pin and - on #4.
No more 162 and 163 errors after power off.

Only had an old 4x AAA battery box lying around to test this. Fits nicely inside the empty fan holder though.

Apologies - those are AA batteries, not AAA.