I've managed to craft some temporary clips out of light weight aluminum. It won't do for holding together a light, but it definitely proves the concept.
It's a little tough to make out what's going on, but the black wire is the ground and the red wire is pulling +4V from the top port.
, on Flickr
All together, the two 3mm LEDs are pulling down just over 0.01 mA. A third light brings it up to right around 0.01-0.02. My multimeter isn't terribly sensitive though so it could be slightly more. I'm currently using 1K Ohm resistors, plopping in something around 50 Ohms makes the light almost unusably bright and draws down around 0.2 mA.
I'm still looking for a quality image of the clips if anyone can provide them. Also, if you have any suggestions for this project, I'd love to hear them.