I'm not sure I've ever cleaned under the lip plate, either. And I've never noticed discoloration. But then, I apparently live in an area non-conducive to tarnishing.
On your sax-specialist tech--give him a try. My repairman is a trumpet player, but is fully trained on most instruments (I say "most" 'cause I'm not sure he does, say, strings. Or pipe organs. Bagpipes. Gemshorns...
) I've almost always been totally pleased with his work. The one time I wasn't was in adjusting my picc's action, but that's as much because of me being particular rather than a failure on his part. He even created a set of pads from trumpet spit-valve foam for my nasty Artley that ate pads. So your sax-tech might actually be quite good with flutes as well.
I'm glad to read how much you're enjoying the head joint's responsiveness. A good flute really is a joy to play! But you should still contact TJ. They might have a suggestion for cleaning off that tarnish more permanently. You never know 'til you try!