The relay in the HCI must ground the red Starter relay/ignition wire (which is connected to the green wire through the relay) when off, thus the green light showing continuity to ground before the run button is pressed.
Adding a diode to the green wire shouldl either make the starter run all the time or do nothing.
The ignition module wire and the starter relay wire are supposed to be connected at the HCI connector so they should have the same voltage potential before the start button is pressed.
There is a possibility that the ignition module likely needs a higher voltage than the starter relay does to operate and that the HCI is bad and cannot pass a high enough voltage and current at the same time to operate both the starter relay and the ignition module without the run button being pressed.
You can remedy this with a low current draw relay (just like the start relay) use the existing wire going to the ignition module power the coil side of the relay (with the other side of the coil going to ground) and run a fused 12v from the battery to the common side of the switched contacts and connect the normally open side of the switched contacts to the ignition. This should make it work like normal. A good way to do this would be to install a parallel a relay just like the start relay, right next to the start relay and jump (piggyback) the red and green wire from the PDM to the same place on the new relay. Connect the new coil ground (where the green wire from the harness would be) to the black ground wire on the old one, and finally use the last contact (where the black wire would be on the new relay) to power the ignition.
The need for low current draw relays is that our hand control circuit boards can only handle about 1.25 watts (0.25 amp(250mA) at 5 volts or about 0.1 amp at 12v). Typically 12 volt relays like the one in the PDM that operates the solenoid, draws about 1.8 watts (0.15 amp at 12v) but has 30 amps capacity contacts. The HCI circuit board probably has the same max current as the hand controls. Typically for a 12 volt relay that has less than 1.25watt current draw the contacts are only rated for about 10 amps max. These are usually circuit board relays (but not micro relays).
Or you could just power the ignition from the running lights (not the headlight, but any other wire that is hot only when the key is on but does not turn off when you hit the start button) but then you have to turn the key off to stop the motor.
If the green wire from the PDM is hot only when the key is on then you could power the ignition from that wire.
Another option is to have the HCI repaired locally (if BDM can repair it, thens so can someone else), you know which wire is having an output problem and which wire feeds it, so it should be easy to replace that micro relay on the circuit board.