The first time I tried this, I only ever attacked Vek'lor and put my trees on Vek'nilash when I could. After the boss died, I noticed Vek'nilash was unlootable, and I assumed that it could have been related to me never actually dealing damage to him so he never truly "tapped" to me (similar to what would happen if I killed a mob with just my trees, they only die and are unlootable).
The second time I soloed them, I went bear first and performed a quick series of thrash/lacerates on Vek'nilash to ensure that I did damage to him that would get him on my aggro table. Then I swapped to Vek'lor and nuked him down through the heals. This way took a lot longer (due to not having the lead up time before the healing began), but ultimately, I did kill them. However, when the kill was over, again only Vek'lor was lootable, and Vek'nilash was not.
Steps to Reproduce:
1. Fight Twin Emperors, focus all spell damage on Vek'lor. Don't attack Vek'nilash (probably need a pure caster for this).
2. Kill them through damage to Vek'lor
3. Observe that Vek'nilash is unlootable.