Matty, the Green Mountain speakers are definitely NOT the problem. While I wouldn't call them bright, the word "engaging" comes to mind.
Also, zero feedback tube amp are more forward than those with feedback, in my experience. The differences are not even close. Right now, my current amps allow one to dial in increasing amounts of feedback, and adding feedback only serves to destroy the sound by taking away extension at both frequency ends and removing all of the music's excitement.
Tube amps with no feedback are also, in my experience, those most reflective of the sonic signatures of their tubes. By applying feedback, you get away from that. In your case, you can use it to your benefit.
Your preamp is known for being on the exciting side, so I would still first change out the Bugle Boys for something more extended. Try Siemens 6922, as they are on the opposite end of the spectrum from the Bugle Boys.
After that, I would roll the tubes in the SixPacs, beginning with the EL84, then moving to the output tubes. One thing I would worry about in going to tubes other than EL34 is the issue many have faced in doing the same thing with the Cary V12. Surprisingly, many here who have tried KT88 have noticed less excitement, jump, power, and bass when moving from the EL34. Despite their speaking to Cary regarding the bias settings, I am 100% sure that the bias settings for these larger tubes is set too low based on the sound.
One EL34 flavor which is more bright than you have now are the JJ E34L tubes. I have a quad of blue E34L coming today, so I maybe able to give you a report on how they compare to my JJ KT88.
But, and this requires a big leap of faith, and the ability to do it, the biggest improvement I have been able to wrest from my system this year was REMOVING THE PREAMP altogether! Running CD direct, straight into my tube power amps increased the jump factor more than I can ever convey. It is one of the biggest improvements I have EVER seen. I was running a Blue Circle BC3, and removing it ratcheted up the excitement, immediacy, slam, bass, power, everything. In later trying a Joule LA150, which was clearly superior to the Blue Circle, the sound was still no match whatsoever for having no preamp there.
The question is, can you do this? Do you spin vinyl? Do you run more than one source? Do you have a CD player with variable out? Are you willing to try one? What about using something like the EVS Ultimate Attenuators? While it is simple, it isn't for everyone...