I don't know the startup current draw for these modules, so I am hopping other forum members can comment on that. What I do know is that what you are experiencing does sound "similar" to the problem with incandescent bulbs and protected cells.
In my PhD-M6 project (link in my signature) I am developing a regulated driver for the SureFire M6 incandescent flashlight, and I can tell you first hand that some of the cells' built in protection circuitry can and do shutdown due to the high amperage, even though the high amperage condition is "brief" in duration. Specifically, I have found that the older blue-label AW protected 17670 cells can't give the 9Amp peak currents that my driver demands when using the MN-21 bulb, but the newer black-label AW 17670 protected cells "do" manage those high currents. The difference between the two cells is mostly the protection circuitry.
In your design you are telling us that the cells are not protected, but that you are using a protection circuitry in front of them. My guess is that your BS setup is experiencing the same situation in which (although brief) the starting current might be just high enough to trip the protection circuitry.
A simple test to verify (or not) my theory is to simply bypass the protection circuitry completely, and tie your BS setup directly to the cells. If the BS light comes ON, then you have verified what the problem is. If the BS lights does not, then we might have to look elsewhere for another theory/idea.
Will
