Can you remove the momentary switch from the dashboard and see how it is wired. It is possibly just creating a open circuit in the wire that runs from the ignition switch to the starter. The ignition switch would still control the power on and off. Remove the switch and connect the two wires from the kill switch together.
But I don't know if I understand how the switch is "always trying to kill the engine". Does it make it hard to get your truck started? Or does it kill the engine while you are driving? If the latter, then my explanation how your kill switch works is not correct.