Tahmid
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I observed in a similar case, when opened the unit to figure out the minimum voltage that can support, there was 7805 regulator inside for the main circuit although supply requirement labeled was 12V. So it could operate down to 8V.
If that is also the case here, all the OP needs to do is connect the battery directly to the modem, maybe use a diode in the path and add a capacitor. That may also be unnecessary depending on the length of the wire from the battery to the modem. There isn't a need to use 7808. Connect a 1N540X diode and you're good for anything upto 2.5-3A.
Hope this helps.
Tahmid.