I've been playing with a KX2 on li-ion quite a bit recently. It appears to be a straight buck driver. Once the supply voltage drops below the Vf plus some overhead, it goes into straight direct drive. Since the working voltage of a single li-ion cell is awfully close to this point, it spends a lot of time there (in DD).
If you can figure out how much resistance you need to drop the supply voltage into direct drive territory, you might be able to get it to work. However, I found that with mine even when you are down to ~3.6 volts it's still pulling about 10 mA from the cell. It's actually fairly bright at this level, but the output is beginning to flicker.
I have a bad feeling that this circuit just doesn't run well at all at the low current levels you would need to achieve a satisfyingly dim level of output.