Generally you want the DC supply to keep the battery pack charged.
For example, have a look at this design idea:
DIY - 12V Mini UPS -- Sorin - DIY Electrical Nerd
https://youtu.be/IlEnEn7ptmcSchematic available here:
It uses a BMS that automatically starts supplying power from the batteries when the charging power supply is removed. The diagram looks like this:
┌───────┐
│ │
│Li-ion │
│ │
└───┬───┘
│
┌───────┐ ┌───┴───┐
│ │ │ │
│ 8.4V ├──┬──┤ BMS │
│ │ │ │ │
└───────┘ │ └───────┘
│
│
│ ┌───────┐
│ │ │
└──┤ DC/DC ├────OUT 12V
│ │
└───────┘