I had some time on my hands so finally got around to trying a few things out to see if they'd work as I suspected they would.
Indeed they did!
First up is my beloved Harman Kardon F500x which a buddy generously gave me. It just didn't work out very well in the main system but when hooked up to a Woo Audio 6 headphone amp it's just right. This was a good match up with Beyerdynamic DT880 headphones.

Where the H-K used to be is now a Sherwood S3000-III which was a big improvement over the F500x in that slot. The Sherwood has a much warmer sound and throws a big soundstage. More gain, too.

I'd tried another H-K tuner upstairs which just didn't work out very well but a Sherwood S3000-V mated with a Jolida headphone amp proved to be the ticket. The Jolida really tamed the AGK 702 headphones, too.

Last but not least, my favorite matchup of all, the El Cheap-0 Harman Kardon PM640 integrated paired with my ancient Stax Sigma headphones. This unlikely coupling turned out to be just great. I was ready to chuck the Stax on Audigon and be rid of them until I stumbled upon this pairing.

There's no real point to this thread other than components don't exist in a vacuum - you need to mix and match a bit until it all comes together.