Can lead acid battery be used as a capacitor between power supply and equipment?
Well, sort of, but not really.


The power supply would need to be adjusted to put out about 13.7 volts. This is the nominal float voltage for lead-acids, and is the point you should aim for.
There would need to be some sort of isolation between the battery and the supply when power is off, or the battery may back-feed the power supply, and may well damage it.
After a prolonged outage, a mostly-discharged battery requires something other than a simple voltage charge, and this would be complicated by the router current load. Go Google on battery charge algorithms.
For a prolonged outage, it's a really good idea to have a way to cut off the router when the battery charge level gets low. It doesn't take a whole lot of complete discharges to permanently damage a lead-acid.


All of these problems can certainly be addressed, and routinely are. The result is called a UPS. Trying to roll your own without knowing what you're doing is a really good way to damage one or more of your components.