XS-3868 module stops noise for 5 seconds, how to make it longer
Sounds like a ground loops and or a missing output capacitor. When the bluetooth module stops playing for x seconds, it turns off its output, so it stops actively driving the output jack. So the line starts floating and picks up the noise from the charging module. You can try a 10 uF ceramic capacitor in line/series on each of the left and right output, to block any DC voltage component on the line.

Secondly try a different charger. Some chargers are more noisy than others and you are picking up the AC or switching frequency noise from the AC to USB/5v adapter.