Combine 1 connector for in and output
You want to drive your LED with PNP transistor, see this answer:
PNP transistor to command LED

Once you switch to this method, you will notice that pin 2 of both LED and charger circuit is connected to ground, while pin 1 always contains signal. Thuse, you can just parallel the outputs. You will want to additionally add a regular diode in series with LED resistor to ensure that charging voltage will not back-feed into +BATT via the transistor.

(edit): Vorstendom provided a schematic: