3rd ign?

the bypass your using should have a (-) status input.

if it does, connect taht status input to the rs160 staus output or ground when running wire.

I'm believe that you are using the DEI 20402 as a transponder.  You are placing a working key into the 20402.  Follow the instructions for the "Transponder" type install starting on page 14.  There are only 2 wires to attach.  The YELLOW /GREEN from the 20402 goes to +12 Volts.  The WHITE/ Green status wire goes to the remote start.  I am not familiar with the IC Dynamics rs 160 but check the install instructions and wiring diagram for a wire called "Ground When Running" or (-) Status Out or (-) 3rd Ignition.  That is where you will attach the 20402's WHITE/ Green wire.  The antenna loop is then wrapped around the ignition cylinder in front of the cars transponder antenna ring.

To disassemble the Taurus steering column and ignition switch cylinder see instructions/pictures at Bulldog Security and look up your car.

http://www.bulldogsecurity.com/bdnew/vehiclewiringdiagrams.asp

I found the RS 160 install manual. 

Attach the 20402's WHITE/ Green wire to the RS 160's Black wire on the 3 Pin connector.  You could also use the WHITE/ Violet wire at the same 3 pin connector if you are not using it for anything else.