Fire loops can be such a pia to troubleshoot at times but it's possible that the connector that leads up to the loop is breaking down when it gets warm. I'm not so sure it's an avionics issue since it turns off when you shut off the rt bleed air and idle back the engine, has it been checked on a ground run or is that even possible? I've only ever worked on heavies so I'm not schooled up on these particular airframes.
