Sounds like toast to me.

Not sure why you want to use a battery... Lets assume a car battery, the voltage of a fully charged 12 volt battery is around 13.7 volts. As you use it, it will decrease to some point where a car will not start say 8-10 volts. And if you started the motor to charge the battery, the voltage will go over 14 volts. I would not think any electronic device such as a router would tolerate that unless it was designed to do so. If you have to use a 12 volt battery, I would think a 12 volt DC to 110 volt AC inverter connected to the battery and the router's power adapter plugged into the inverter would give you more stable power. Yes the power consumption of the battery would be higher. Having said that, I am not an electrical engineer or such and no guarantee that it would work. Just my two cents.

I live in the woods, off grid, and am attempting to stretch my little solar system by using a single 12V deep cycle battery, charged by a 120 watt solar panel to power my modem (this is done and works great!) and router. I was a bit misled by a friend who said if I got the polarity wrong on the router wires, it just wouldn't work. Well, it not only didn't work but it toasted my router! The good news is that by doing this project myself, I've already saved Big$, so having to spend another $70 on another router is not the worst thing. Yet, my pride suffered since I could have referenced a 1minute YouTube video that showed how to easily determine the polarity of unmarked wires! Live and learn! (the new router WILL NOT be plugged into the battery!)