Since the charger point and battery points are the same P+ and P- terminals on BMS board. so when there is no charger battery will supply the power to modem.
I don't think so. Look closer at the board. There is no direct connection from P- to B-.
If there was a direct connection, what would this board even do? It would have no way to disconnect the charger, drop the voltage, regulate the current. It would be useless!
I had a look at some ebay listings for this board, which had information ranging from okay to spectacularly useless. None clearly said what the maximum voltage input on this BMS is. It could be 13.6V.
If you have 13.6V on the P terminals and the batteries were almost fully charged, the BMS would be dropping 1 volt between its input and output.
How do you know this BMS uses TP4056s? It doesn't say anywhere that I could find.
In conclusion:
You should attach a DC power supply to P+ and P-. It doesn't control the charging cycle.
Modem should be connected to the batteries.