do you think the Lyr has a good soundstage? It would be silly to get these headphones with a great soundstage and positioning and then ruin it with an amp....
If you want the best sound stage I think the Lyr2 would be decent, my impression is the sound stage of the Lyr2 is about as good as it gets for anything under $500, its better than the Ember and BH Crack to me in sound stage but the overall sound is better with the BH Crack or Ember, to me the Lyr2 is not a great pairing with the K7XX. I know with the AudioGD Master 11 the sound stage and overall sound is about as good has I've heard with the K7XX, but its a $2K DAC/amp that will drive anything, including an HE-6, its at 8W @ 50ohms, overkill and very much a diminishing ROI.
If you really want sound stage I would look at other headphones, might be cheaper versus trying to find a DAC/amp that will preserve the sound stage. Its why I went to the HD-700 for gaming and watching movies, the sound stage is better than the K7XX and its easier to drive IMO. I think even the DT-990 premium as a better sound stage than the K7XX, to me anyway. The HD-800 has a very wide sound stage but with some genres with lots of vocals it sounds off, vocals done much better with the HE-560 which also has a good sound stage.