It might be time for a visit to the vet, Katie. When was his last blood panel done?
This case really does mystify me, though. Why every 3 weeks? As Laura said, something that gets rotated in at a rate resembling a three week cycle is the most likely culprit, (and not even necessarily on the same week. Something from the week before could take a week to reach "critical mass", so to speak) but seeing as how you've looked exhaustively into that possibility, we have to start considering options a little more out of the box.
Maybe the link to the food is more indirect. Do you shop for all or your foods at the same stores? Is it possible that there is a store that you buy from roughly every 2 or 3 weeks?
Any cleansing agents used around your home on that cycle? Lotions? Air fresheners? Perfumes? Just spitballing here, in case something sticks. How about when you clean the cage or any of his perches? What do you use?
I think you have to examine anything that you do that adheres to a fairly regular routine. And then beyond that, to anything in your home that occurs at fairly regular intervals.
Oh. Question. When this happens every 3 weeks, how long does it last? Is it one prolonged occurrence? Or several? Taking place over one day? Or a few?