Ok, here'* the deal. I think (I'd have to check) that the monitor watches each individual light. I'd suggest putting one stock light back in, and playing with the other. Get a high-wattage potentiometer or rheostat from any electronics supply store. What you're going to try to do here is either increase or decrease the current draw depending on what you think the problem is. To increase the draw, install the pot across the hot and neutral leads of the headlight.....Use a 1meg or 10 meg pot. On it'* highest setting, it might be enough already. To decrease the draw, you'll need to install it on the wire going back to the lamp monitor, and the other end to ground. You'll 'bleed' some current off to ground this way on a parallel circuit. WARNING. My schematic is not very detailed. Block diagram for the most part. Somebody with a set of factory service manuals will have to step in here to verify this until I can get my hands on a set. If this is something you'd really like to try, I can dig further for you.
It'll be alot easier for a lamp that isn't drawing enough current, like if you switched to the LED 1157 taillights.