run ac device on battery
The radio will internally work on DC. If faced with the challenge I would open it up, find where the AC to DC conversion is done and, taking great care with any exposed live wiring or terminals, measure the internal DC voltage. Then I would attempt to take a current reading with the radio playing at the required volume. Since current readings are taken in series with the load it may be necessary to temporarily disconnect a wire.

Based on this information the battery voltage and current requirements can be determined. The mAh (milliamp-hour) specification of the batteries can be calculated by current x run-hours x safety factor. So for example, 650 mA x 2 hour x 150% = 1950 mAh or 2 Ah capacity.

Next you have all the trouble of fitting and connecting all of this, making some way to switch from mains to battery and some means of recharging the battery.

Worth the trouble? I doubt it.