Electrical – FS8205A won’t kick in

Those modules protect against over-charge, over-discharge and over-current, but they do not have a balancing function. Cells that are matched and charged to the same voltage initially should not need balancing. However if you are worried that your cells that might go out of balance then you can use a PCM with balancing such as this:-

4 cells 8A w/balancing Li-ion Lithium 18650 14450 Battery Input Output Protection

The balancers in this PCM bypass up to 42mA from each cell as it becomes fully charged. To properly charge the battery you still need a Li-ion charger or power supply which provides constant charging current and limits the peak battery voltage to precisely 16.8V.

Cells in devices that you do not wish to be a flaming ruin MUST have protection.
Having protection circuits does not guarantee it will not happen.

Cells without protection are intended for use by either manufacturers or experts or enthusiasts who add their own or for suckers.

Whole device protection and cell protection are complementary and serve overlapping but different roles.

4,000 mAh 18650 LiIon batteries are ~~~= 99.9% +0.1% - 0.0% sure to be rubbish.
ie not just < 4000 mAh but << 4000 mAh and low quality.
The people who bogus label cells

• almost never feel an obligation to use the best cells they can and

• almost always decide to add injury to insult by using junk as well as lying.

Real world experience shows that the value of 'almost' is very high in both cases.

Higher capacity cells can usually be fitted OK.
MUCH higher capacity will lead to long CC tails and overcharge but not an issue here as mAh_new is < to << 4000 mAh.

ADDED:

Notice that in this ad and all their other ads they ALWAYS show non-brand-label views of the battery.

However, you may find that the racing stripes and general colour scheme a good match for the well known "Ultrafire" brand batteries. This may in fact be a real brand and these may be real examples of it BUT you can buy empty shrink wrappers to apply to the battery of your choice with this (or other) branding on it , so caveat emptor. Better nullius emptor I'd hazard.

These ones are a stunning 6800 mAh - a steal at the price. Available here

You'll find others similarly arrayed here and
here - 3000 mAh and 4000 mAh and

6800 mAh !!!!!!!!!!!!!!!! {again}

and unspecified but with GENUINE CREE 2000 lumen {so 20+ Watt} flashlight for $9.27 and
only 4000 mAh and
that's better - 4200 mAh
and .............

Flee!