I got under there today and sure enough, the small heatshield right below the driver's seat was the culprit - one of its weld joints had rusted through. I bent it up so that the edge was an inch or so away from the pipe and the sound went away.

Any suggestions on how to properly fix it short of getting a new one welded on? Is there some sort of high-temp adhesive I can use to reattach that joint?