Balanced will allow for 4x the power, yes. Unbalanced gives you 62.5mW and Balanced gives you 250mW
I've only had the HD820 for roughly a month, not sure how much different it is to the HD800S but i could drive the HD820 perfectly on Low-Gain with the WM1Z and the M11 Plus. For very quiet songs i had to go to Mid-Gain on the M11 Plus but because Low-Gain on the WM1Z is louder than Low-Gain on the M11 Plus.
But other than that, i did not have any issues. You will, of course, be able to speeze a bit more sound quality out with an Desktop Amp, but that is always the case, no matter what Earhpone/Headphone you choose.
At lower voluems, the DAPs even performed better than my Desktop Amp. Capacitors change their sound depending on how much power you drain out of them. If you listen at lower volumes, Desktop Amps often start to sound meh.
But the HD820 were very easy to drive, i was surprised. So if the HD800S are the same, you won't have any issues