The pack and charging method should work fine, however, there is another problem that has been overlooked, allow me to elaborate...
The MTE 2-mode direct drives the P7 on high (as I understand it, correct if I'm wrong here), and it is more by happenstance that a single 18650 happens to drive one well, without overdriving, ...because the single cell experiences an ideal amount of voltage sag to come to a sort-of ideal "happy medium" with the P7 for reasonable drive levels. My concern is that by piling 4 cells in parallel, the load split between the cells will allow the cells to hold a higher voltage, this could be the difference between pushing the P7 at a "normal" ~2-3A range, and maybe more like 4-5A+, which would probably end up tripping your PCBs high current limit, or, if you had a PCB with much higher current limit, would just end up overheating the P7 rather quickly.
I'm under the impression that the reason for putting 4 cells in parallel on this would be to extend useful runtime, but I'm concerned that in doing so, you'll actually end up creating a thermal problem that limits useful runtime substantially, and potentially damages the LED prematurely.
This is not to say that this project can not be done! You might just need to add a regulator into the circuit! Search for some of the driver solutions people are using to push P7s around here for some ideas.
Eric