This one's gonna be tough.
Start with the exhaust, metal expands when heated, but you knew that already, with that in mind, talk it for a ride and when the noise starts take it to the house and jack up the rear end, use jack stands of course, we can't have members get'n squished, start it up and put it in gear see if it rattles, check exhaust connections and hangers, then the drive shaft, trans mount things like that.
Let us know what you find or don't find.