Although, the Little dot MkIII does possibly not sound like the ideal partner for the HE-500s (and surely there is a lot of amp headroom that can be filled), I think it can be good with the HE-500. You may want to set the gain to +10, although +5 is useful too. As power tubes, I would try Novosibirsk 6N6P-IR or if you have more change Sovtek 6H30 type tubes. 6N6P-IR have a bit more bass, 6H30 better treble extension and separation. As driver tube, I would suggest Mullard M8100 with a small Mullard logo. I am currently listening to this setup at 10 to 11 on the volume knob and it sounds loud enough, pretty nice and authoritative. This is with the HE-500 having been used for only like 15 hours.
However, I have some very nice Tung Sol driver tubes (JAN-CTL-6AJ5), which just do not pull enough for the HE-500. They are predecessors of the 6AK5-type driver tubes and hence provide lesser voltage.
As I have only read vague and not necessarily good comments on the LD MKIII + HE-500 combination, I wanted to add another point of view.