TPR0 wroteI would be concerned about brownout, and battery service life(should not drop below 10V , or sometimes 11)
Just get a digital voltmeter gauge and monitor or a cut off circuit, as the power system in routers uses smps, would heat up and die if its running under operating voltage by a large margin.
Be AWARE of how you are powering the devices directly, for example, running the router directly from the battery while the ups is plugged in to mains interferes the charging circuitry of the battery, e.g: battery won't reach 13.7. Or, certain line interactive ups (use transformer as a regulator) have the primary center tap connected to the battery, few spikes would reduce the router's life easily.
Just get a basic 12v 7-14ah battery charger+ battery and use your diy skills, like a simple relay switching circuit with its own 12power supply according to your router's specs.
It took me months to figure out that all I needed was a simple relay. I'm stopping by Ketranji and getting one today. The old setup didn't work as expected, I had o manually manually switch between the battery and AC adapter since the battery's "smart" charger is getting confused while powering both the modem and battery. So when they were connected like that, the battery drained and the modem shutdown even though it was connected to the mains at all times.
I'll keep you guys updated!