Some batteries have no fuses, some have a charge and discharge fuse, some have just a charge fuse and some only a discharge one.
You only benefit from a charge fuse if there isn't one already in the charger.
A discharge fuse is a bit of a waste of time because it has to be such a high value that the wires will melt before the fuse blows and the discharge from the BMS is protected in the case of a dead short.
Some people like to err on the side of safety: They won't go outside in a thunderstorm in case they get struck by lightning. Other people won't fly on an aeroplane in case it crashes. It's a personal choice.
The only thing OP doesn't understand at present is the flashing green LED on the charger, which might mean everything is normal. That problem (if you call it a problem) is completely independent of fuses.