Using 12V 7ah battery in parallel as a cheap UPS

[Kneerly Down]

Due to our unreliable power provision and the fact the winds are supposed to be getting quite high shortly I was looking at ways to keep the internet on.
We've now got fibre to the outbuilding that is being fed by a long UTP cable to the router that distributes the internet in the house.
The router was co-located with the old DSL modem so both were on a UPS.

Rather than get yet another UPS that takes 12V battery up to 240Vac and then back down to 12V I wondered about getting a slightly chunkier regulated 13.5V PSU and put a standard 12V 7Ah VRLA battery in parallel so it is kept charged and when the power drops it just keeps supplying power to the router. The battery (or batteries - but need to make sure charge current when the power goes back on doesn't do for the PSU) will last longer than in the UPS as it's not going through an inverter then PSU to power the router. Not sure at what voltage the router would stop trying to take power from the battery when the power is off either.

Anyone tried/seen this type of setup?