As for running routers off 12v, you skip the inverter altogether. Most of them use a 12v DC wall wart, so you cut off the 110v plug part wire the cord straight to the battery, then add a $4 HF trickle charger and you have a UPS that'll run your router for quite a while if the power goes down.
Just some comments -
1) ATT U-verse here at the ranch, and at my home in town. The one in town goes down for a moment several times a day, many service calls couldn't fix it. They told me it was my power, or bad ground. So a U-verse tech gave me a 12v backup battery and I tried running from that as a diagnostic tool. Didn't help but they said I could keep the battery.
ATT is now running fiber there and sent me notice I could switch, at additional cost of course. I've read ATT is getting out of the copper-to-home business entirely, will sell remaining customers to small home town providers. My suspicion is my signal is bad due to total neglect of the 100 year old copper infrastructure, or possibly intentional bad service to drive off the last copper customers.
2) I tried that HF $4.99 trickle charger to maintain a tractor I didn't use for months at a time, concluded that leaving it connected past a couple of days boiled out the battery water needing water added occasionally to a battery that never needed this before. I attach it now only if a voltmeter shows around 12.4 volts or below, and only for a day or so. For continual use like internet service I would buy a more sophisticated charger.
3) I have a jump-start battery here at the ranch, and have the connectors to use its 12v for the router, 12v phone chargers, and some LED lighting, if needed. Also an inverter if needed for some other gadget. This battery is a little neater than a home brew system built up from a car battery.
4) And sorta relevant: I read some techie wired his Prius to power a fairly large inverter for minor emergency lighting etc in his house. Not using the high voltage that drives the traction motor, but rather he said because the engine is frequently restarted a Prius has a huge (120 amp?) 12v alternator and large battery ideal to power an inverter. He said the car starts itself and runs a few minutes to recharge its 12v battery, at something like hour intervals. He said he looked into tapping the high voltage/high amperage propulsion battery system and that looked to be complex, expensive, and dangerous, not worth the trouble for something he would rarely need.