I'm fairly certain your pack's a mis-named 11.1v nominal pack using three 3.7v nominal 18650 batteries (full charge is about 12-12.6v, but batteries aren't supposed to be rated at their topped-up voltage). That probably explains why the second indicator is off by about 15% - you'd probably have better accuracy with an 11.1v lithium-ion or lithium-polymer voltage checker. Something like this perhaps: https://smile.amazon.com/dp/B06Y1HVR53/
As far as I can tell, the battery pack's protection circuit was left inside the pack's old case to use as a charging adapter. That should help balance things during charging (since the voltages are probably a bit uneven splitting off the two cells for 7-8v to the 5v regulator), but that still leaves you with three unprotected lithium-ion cells inside your helmet, which is a safety concern.
If you want to keep the batteries in the helmet, I'd suggest permanently wiring the protection circuit back to the cells and only using them with the full 11-12v line. Your 5v regulator looks to have a 25v max, so you're fine feeding it more voltage, though it might give off more heat.
Alternatively, if you can find some way to insulate these from moisture, a 3x18650 holder (like this: https://smile.amazon.com/dp/B071ZQ2X2T/ ) and three *protected* 18650 cells would give you the same amount of power in the same amount of space while also giving you the option to swap batteries if you happen to run down mid-troop. If you want to cut down on space/weight, you could also use a 3xAA battery holder with three protected 14500 cells.