Knigh7s, that is what the diagram entails:

Key out; rear popper works by either pressing the unlock button on the stock Isuzu remote twice, or pressing the unlock button on either door.

Key in (and on); rear popper works by pressing the tailgate release button

Depending on the load that your new external button can handle, you may need a relay no matter what. You could do another five post relay that has constant power when the alarm isn't on, but when the alarm is activated it would throw the relay and cut power to the external button...... I just have no idea how to achieve that.

Like etl said, you would need to find some constant on power coming from the alarm system somewhere.