The module that drives the low beams is not like a standard relay such as run the fuel pump or the high beams or what have you. This is a solid state "driver" module. The headlight switch doesn't really turn it 'on' either.
What happens is this... That driver module is supplied with a 12 volt positive and a ground all the time..Key on, key off, truck running or not, locked up and parked doesn't matter. That driver always has power. Now if you remove an original HDM (headlight driver module) and look at the side of it you will see a little circuit drawn there like this...
The 12 volt positive I just spoke of is connected to terminal #2 and the ground is at terminal #6.
The headlights are connected to terminal #4. That leaves terminal #5 and that is the control that is connected to the BCM.
Now see that line going from #2 to #5 with the squiqqly line over by the #5? That squiqqly item is a resistor and in this instance we call that a pull-up resistor because it is getting power from that #2 terminal with 12 volts positive and 'pulling up' the #5 terminal voltage to 12 volts.
Now then that terminal #5 is wired back to the BCM and when the BCM wants the low beams on then the BCM kind of grounds out that 12 volt signal and that drops the power at terminal #5 down lower.
That HDM module is built such that if the voltage level on terminal #5 is high, like 12 volts then it doesn't send power out to the low beam lights. But if the voltage on terminal #5 goes low, closer to zero then it sends power out to the low beam lights.
Why not just use a normal relay like we see on the high beams? Because they wanted to be able to run the low beams duting the day as a safety measure but they didn't want to be using full power for that.
So during the daytime when the BCM wants to run "daytime running lights' it grounds out that terminal #5 control wire but not fully. The PCM uses a pulse width modulated (PWM) signal on that control wire and this makes the HDM send out about 80% power to the low beams.
So how does this help you? Well if you followed all that blather you can see that all that has to happen for your low beams to turn on is that wire from the HDM #5 back to the BCM gets grounded out somewhere and BAM!! , Your low beams come on.
Now you say you unplugged the BCM. Just how did you do this?? If you unplugged the connector #1 with the lo beam control wire and the low beams stayed on then somewhere along the way from that HDM module and the BCM the wire is shorted to ground.