Qubeet
6
3.6 (or 3.7) is a nominal voltage of Li-Ion battery (specifically NMC). Actual voltage changes between 2.5 and 4.2 volts (0% - 100%). So 4s pack will have 16.8 volts which is too much. You should build a 3s pack.
Since the jacket works fine with an automotive voltage range you can look at LiFePO4 batteries. They have a nominal voltage 3.2V and actual range 2.3V-3.65V. So 4s pack will have 9V-14.6V.
LFP batteries require specific BMSes and chargers.