Just to recap a bit of this that we know so far:
The H6 can carry 6 pounds and weighs 2.9 pounds in aerial mode.
The H3 can carry 3.7 pounds and weighs 1.9 pounds in aerial mode.
A GH4 with 12mm f/2 Oly lens is 1.5 lbs. The Blackmagic pocket with the same 12mm f/2 Oly lens is 1.1 lbs. The new micro cinema is a couple of ounces lighter yet. The Blackmagic cinema (the bigger, yet still compact option) is about 4 lbs with that same 12mm Oly lens.
The Sony A7s/r is about 1.3 lbs with a 35mm f/2.8 lens.
The Canon 5DIII is about 2.7 lbs with a 24mm f/2.8 lens.
So, the H3 can handle all of these options apart from the larger BMCC. Yet, the H6 is only 1 pound heavier in the air and can carry everything above and then some.
For more reference (and just because the specs are handy), DJI's big octo -- the S1000+ has a payload of about 14 pounds. Drop about 5 lbs for a big battery (or pair) and electronics, and you have around 9 pounds of capacity left over. That can haul the H6 at 2.9 lbs and its max payload of 6 pounds almost exactly. That would be the top end, though all of the camera options I list above would be under the specified max operating weight with the H6 by at least a couple of pounds. Naturally one is not limited to the S1000, but I just mention it here to give some idea of what might work.
Tough call. The H6 offers more and has a full cage. The H3 is lighter by a pound, but may not be as robust (hard to say on paper). Apart from physical dimensions and capacity the H3 and H6 appear to be otherwise comparable (same software, etc., as far as I can tell).