Based on the image of the KD9910, I don't see how it can deliver the claimed 1A output. It looks to me that the sense resistor is 0.47 ohms, giving 425mA output.
Let me explain something that you may not realize or understand. There is no standard method for specifying the voltage range for a driver. The upper limit is often based on the maximum rating of the buck, boost, boost-buck, or linear regulator IC that is used. The lower limit is typically the iffy figure. Is it the minimum voltage required to run in regulation? Is it the minimum rating of the buck, boost, boost-buck, or linear regulator IC? Is it the voltage for initial LED turn-on (and which LED -- a new XP-G, an old XR-E, a Seoul P4,...?). Is it the voltage that gives 50% of the rated output current (again, which LED is used when determining this)? Something else? Who knows?
So when KD9910 claims a voltage range of 3V to 12V, the whole thing is somewhat iffy. The PT4105 buck IC has a recommended operating voltage range of 5V to 18V. It has an absolute max rating of 20V. Certain components (a resistor and a capacitor) are recommended to be added to the typical application circuit shown in the PT4105 datasheet if you operate above 12V.
As best as I can tell (I'd need to trace the connections on the board to make sure), I see the recommended 1 ohm resistor, but I don't see the capacitor. That might be why KD specs the board only up to 12V.
But what is the 3V lower limit? Can't tell you that. I can tell you that for the drop-ins that I have which also use the PT4105 at 1A output, they all need about 5V input to get the driver to run in full regulation (LEDs are a Cree XR-E P4 and an XR-E R2). At 3V, I measured about 100mA drive current (i.e., a dim LED). I suspect that's why the PT4105 datasheet has a recommended voltage range starting at 5V, not 3V.
With the KD1640 driver I suggested before, and driving an XP-G R4, I measured an input voltage of about 4V for the driver to run in full regulation. My driver/LED pair required about 0.7V above the LED's Vf to run in regulation. Others have measured even lower required voltage.
This post claims 0.5V voltage overhead.
Basically, 1x18650 will probably start in regulation, but probably stay there only for a short time as the battery drains. At an input voltage under load of say 3.8V, I would estimate that the drive current will fall to 500mA or less. The KD1640 probably won't run when the voltage hits about 3.3V to 3.4V (your Li-ions would essentially be empty anyway at that point).
That's the challenge. If you want to have good run time in regulation with 1x18650, I'd suggest using a 3xAMC7135 based driver (single mode or multimode). If you want to run with 2x16340, then the KD1640 will work well. If you want the flexibility to run either battery option, then I don't know of a driver that can run well in regulation for both cases (unless your KD1640 sample has a lower voltage overhead than the one I got -- if you get one with an 0.5V overhead, then you are probably in good shape to use either 1x18650 or 2x16340; but you can't tell just by visual inspection).
