What router? 12v5a is 60w max, which is not an unreasonable amount of power. A 1500va UPS should power it for like two hours or more.

You could easily wire it up to something like that. Take the power cable, cut off the wall wart, connect the positive wire (usually red, could be white) to the positive terminal, the negative to the ground terminal. Cut off the head of a power lead, connect the live, neutral, and ground to their terminals on the AC side. Plug your battery in, the power lead into the wall, and you’re done. Check everything with your multimeter before actually wiring it up if you’re not sure. You do have one and know how to use it, right?