Hi Bjoern, Welcome to the forums. I'm sorry no one has addressed your question sooner. This is kind of odd as there are quite a few knowledgeable people here.
Personally, I have tried looking through the website that you linked to, but there aren't alot of descriptions for the light that you mention. In fact, I'm not even sure that I've found it. Could you supply a model number, or the precise link to the product?
You mentioned that you tested the LED. You mean to say you tested directly across the terminals of the LED, or did you test across the battery output terminals? Are you able to supply photos of the lamp and the LED?
Barring any further information, there is not much I can tell you. All I know is that a proper white LED requires about 3.6V before it will light up, though the current it draws can be less than the stated 30mA. To overcome this limitation, companies have come up with various DC-DC convertion circuits, some more efficient than others. Some companies who have done this are:
Arc Flashlight,
CMG Equipment,
Brinkmann LED light and the Princeton Tec Matrix. (There are quite a few others, but I think that is enough for now.
)
Another way to drive the LED is to use the 555 timer IC and a bunch of capacitors which charge and discharge, but this produces a flashing light rather than a continuous beam.
Hope some better answer is forthcoming.
I certainly would like to find out more about low voltage white LEDs.