Courtesy of the TMP boards and the modelling brilliance of Scott Washburn. Bravo Sir! Bravo!
Here is his post in toto:
A little drilling, a little clean-up and it's ready.
I toyed with the idea of making another hole and poking the button of the switch through the body of the tripod, but I decided I would just leave the top of the head loose and access the button and the battery that way. The pegs in the top of the head fit very tightly into the holes in the lower part of the head. I wanted to loosen that up, but not too much. A little careful drilling did the trick.
After that, it was just a matter of hooking the wires of the LED up to the wires of the switch/battery and stuffing the whole thing into the head of the tripod with the light poking through the hole. Oh, I strongly recommend finishing all your painting before mounting the light!