Why not eliminate the inverter and just run the radios from the 12V battery during power outages? That would be the most efficient. If you are not running unattended then you really don't even an automatic switch over.
Alternately you could always power the radios from the battery and use an automatic charger to keep the battery charged when AC power is available. The TX/RX duty cycle is typically low enough that the charger doesn't even need to keep up with transmit current. Power will be taken from the battery during transmit and the battery replenished during receive.
I would still do a large inverter with extra batteries attached to it. I use a 1000 watt one with extra batteries. It uses two 12 volts in series for 24 volts to run inverter. It powers cable modem, wireless router a desktop computer, a loptop and a light . It also powers a SS 30 for rigs. I can run 4 to 6 hours on UPS depending on load. I can power it all at one point with a generator feed which recharges UPS batteries too. As longer as you use switching supplies it is pretty efficent.