Wow! Just found this thread. Super awesome work, man. Well done! GL's my 2nd favorite superhero (only to Batman), but he is the one who got me into reading comics in the first place, with Blackest Night. Your ring is what drew my attention, though. There's a way you can get an equally bright (or maybe slightly brighter) LED that is the same size, but doesn't require nearly as much voltage. You'd only need a 7.4V pack, and with 2 18650 sized cells, you'd have a pack rated at 3400mAh-massive runtime! I actually did this same setup in a lightsaber a while back.
Get this LED:
LED Engin LZ4 10W Single Color
And this CC driver, in the 3A, momentary option (expensive, I know...but ultimately, you will get the same brightness you have now, with a much smaller battery pack, and longer run time.)
Constant current LED driver
Then get this battery holder:
Spring Clip Black Plastic 2 x 3 7V 18650 Pointed Tip Battery Box Case Holder | eBay
And two of these cells:
Panasonic Protected 18650 3.7V 3400mAh
And a momentary switch:
SPST Momentary tactile switch
Then, what you'll do is wire the 4 LED dies in series/parallel, which means take 2 dies, wire them in series (pos to neg). Then take the other two, and wire them in series as well. Then take the two pairs you just made, and wire them in parallel (pos to pos, neg to neg). What this does is split the initial 3A current the driver is giving the LED in half, to 1.5A to each pair of dies, then since each pair is wired up in series, both dies in a pair gets the same current (1.5A) (parallel wiring splits current, but keeps the forward voltage requirement...series keeps current the same, but adds up the forward voltage requirement). So, you have 4 dies, getting 1.5A each, and at that current each die would conservatively put out around 600 lumens, according to the data sheet. And you got 4 of them! Plus, you only need 7.4 V , and have a 3A drain, so you'd get over an hour of runtime.