I understand your frustration. You invested a large amount of capital in your rig - and it's a great rig - and now the sound is a bit too bright for some of your music. Here are a couple of ideas:
Two ideas in which I don't have any personal experience, only Head-Fi comments):
1. After-market mod the HD800. You can find details on the HD800 Appreciation thread. Looks pretty easy - just adding some foam inserts at strategic locations inside the ear cup. There are pictures and a lot of discussion on that thread. I believe these mods are intended to reduce the "brightness" of the base HD800 sound.
2. Like you already suggested, change HP's. Along with the HE560 that you mentioned, others that come to mind are the HD600/650 or the HD700 that's come down quite a bit in price. At the Music City meet, a month ago (there's a thread) there were favorable comments about the HD700 cans being warmer when compared to the HD800. Just note that the output impedance of the HDVA600 is quite high, for a head phone amp, so pairing with higher-impedance cans would make sense.
You can also adjust the gain on the HDVA600, but I'm not sure that will satisfy your need for great sound.
Personally, I have two rigs, both fed from the Gungnir (which I love). Warmer: Mjolnir --> LCD-X/XC. Cooler: HDVA600 --> HD800. I use these for different types of music. The Audez'e sound, in particular the X, is warm when compared to the HD800 while still being very resolving. You can also consider the LCD-2 that is a real bargain at around US$1000. The XC have a warmer sound than the HD800, cooler than the X, and they are closed. If possible, audition all the LCD cans.
Note: The X/XC do not pair well with the HDVA600, IMO, possibly due to their low impedance of 22 ohms. I never listen to my LCD-X/XC with the HDVA600. It's possible that the LCD-2 may pair better with the HDVA600 as they have somewhat higher impedance than the X/XC. However, the HDVA600 was designed to drive the HD800 that have an impedance of 300 ohms.
Hope this helps,
RCBinTN