That DC-DC converter you linked to can only supply 280 milliamps. The E-bay listing for the LEDs Andy was using specified a 400 milliamp current draw.
The other issue is the low capacity. That NL14500A Li-ion is rated at just 650 mAhr. The DC-DC converter will be drawing 1 amp from it. So even if the LED will run on the lower current output from the converter, the battery will only last about 35 minutes before it's dead. Although it would be a very small form factor, I don't think this is enough capacity for what he wants.
Now, three of these 18650 Li-ion batteries wired in series could supply 12 volts at 400 ma for over 7 hours... (Link is for a pack of 2 batteries, but you can search for other deals.) But if you add up the volume of those 3 batteries, it's no smaller than the original 3S LiPo battery I linked to above.
Adam