Seamlessly powering 12v CPU fan with regulating speeds
As the battery is for backup, I assume it's a gel-cell type, so suitable for float charging at a constant voltage.

Buy or make a float battery charger, capable of providing at least the 600mA the fans require, set to around 2.25 to 2.3 volts per cell. A linear regulator would be fine. Make sure the charger can tolerate the input power supply dissappearing without discharging the battery.

Run the fan controllers directly from the float charged battery.