Tried?
Heck, I've punched holes in quite a few perfectly good KL4 PCB's! I think driving the LED at a higher current level than it is will give you more thermal grief than luminous output if you have it on for more than a few moments.
Hopefully CM will be getting a feel for possible tweaking of the existing circuit itself. If it is a "typical" boost circuit, it may be that it can be used as is and power a Luxeon III on a single 123 or one of the Li-Ion cells.
The KL4 reflector is very friendly to the single die of the 1W and no doubt the III as well. I should know more later today, I hope. Since I am ignorant of the existing driver, I will be putting a DB1000 in its place.
- Don