I think I may have figured it out.
I opened up the controller that came with the bridge. It is labeled "Automatic Switch Controller". It appears as if the center connection inside the controller is "common".
There are two heavy copper strips with each of the other wires connected them. My guess is when you make contact via the lever that you complete the circuit and make the bridge go either up or down depending which way you press the lever.
There are three terminals on the front of the bridge where the controller gets connected and two on the back to supply power. The center wire of the controller is connected to the middle terminal (I suspect this is the common) and the other two wires are connected one to the right terminal and the other to the left terminal.
I would power the bridge independently with my K-Line accessory transformer.
So, I could simply wire it up just like the controller that came with the bridge. I currently control power to my yard tracks by using the ASC ACC 9-12. I could use two of the remaining terminals as ACC 13-14 connecting the COMM to the center terminal on the front of the bridge and the wire from #12 to the right terminal and #13 to the left terminal. By addressing the respective ACC and then pressing the AUX1 button on the CAB1L to raise or lower the bridge.
Does this make sense?