Here's something that i tried.
I am not done putting my bike back together... so i did this on the kitchen table.
I took a 9volt battery. Some 12Volt Auto LEDs.....
#1. BLACK wire (on the black remote box labeled "GROUND") and the Ground wire from the LEDs to the 'negative' post on the 9volt battery.
#2 RED wire (on the black remote box, labeled "12 V DC IN") to the 'positive' post of the 9volt battery.
#3 WHITE wire (on the black remote box, labeled "12 V DC OUT") to the hot side of the LEDS.
Using the Keyless remote (similar to the remote for your Car Locks) hit the 'ON' button,, the LEDs light up. Hit the 'OFF' button,, LEDs go off.
I have to put it on my bike,, but it seems to work in testing.
Does anyone know,, how much "draw" is on the Big Dog system,, when the key is ON,, before the bike starts??
If i need to,, i can add,, more LEDs,, to increase the resistance,, if this might be the problem.