exhaust backfire

THANK YOU for listing the year and engine and all the things you have tried, so refreshing.

Just some thoughts...

Since it only happens at idle, when hot, points me in two directions,
  1. An exhost valve that sticks open occasionally but only when hot. Could be a clearance issue, as in when something, (lifter, rocker, valve) heats up it grows and the friction is such that it sticks open just a tad. Could also be a lifter pumping up, heat related
  2. The electronic ignition is malfunctioning when it gets heated up throwing spark at the wrong time.
You mentioned it runs rough at idle at times. Does that happen at the same time as the backfires.