Ok I have looked at this and thought about this some more and here are my comments:
1. I trust you are now confident that your cables are ok, I would suppose you would even have other HPIL devices you could try them with to be sure but just joining the loose ends as I suggested previously is a pretty good indication.
2. I took out and plugged in one of my 82164As and observed that initially both lights are on but after a moment the T/R light goes out and it only flickers where the 82164 is actually sending or receiving so that may be a happy sign that the controller in your box is actually getting off the ground.
3. The flashing power light is probably just indicating low voltage that may or may not be fatal to the operation of the box.
4. I am concerned about that big resistor near the + terminal of the right battery pack I think you one pack may not be charging. Are the two packs connected in series? I would expect that they are as 3 NiCds is only 3.6V nominal, not enough for the logic on board. Did you try checking the power inside? I suspect that nearly everything runs on +5V probably the easiest point to check would be on the EPROM pin28 (+) pin 14 (-) same for the flat pack SRAM next to it but less convenient. I trust that you wired your packs to the board the same way as the originals where connected.
5. Have you looked at the underside of the board? There could be battery leakage damage there too.
6. Continuity on the HPIL connectors with an ohm meter on its lowest scale each pair of pins will look like nearly dead shorted but it is actually in the neighborhood of 0.5 ohms across the in and out pairs because the DC resistance of the windings on the transformer are quite low.
7. On the other side of the transformer between pins 17 and 18 of the ILB3 you should measure about 2 Ohms and each of these pins should measure around 7K0 ohms to ground. Each of these pins has a 15K resistor to ground but because of the low resistance of the transformer winding they are essentially in parallel. Between pins 19 an 20 you should measure around 700 to 760 ohms each of these pins has a 383 ohm resistor between the pin and the transformer. On TOS there is a a Technical Manual for the 82166C that contains the best schematic of the circuitry between the the ILB3 and the HPIL connectors. There are also some zeners and capacitors but they are not really critical unless thy go short.
I suspect you may have a double fault here, something wrong with the power supply and also a fault in the HPIL, unless of course the design is halt operation when there is a power fault such as a low battery.