Umm maybe I'm missing something but isn't the point of battery backup to NOT let the clock fail?? Can't you just run a bigger battery?
For my master clock I have a 12v 45 amp-hour car battery, kept on float charge at 13.8 volts, then the entire system is run from the 12v battery. When we get the (typical) 4 hour blackouts the battery barely dips. My battery is a little sulphated, but it easily supplies 1 amp for 12 hours, and my system is only 100mA or so for master clock and peripherals.
If you separate the actual clock from the peripherals with diodes, (bacause you don't need chimes and line drivers etc in a blackout), using the battery to preserve just the clock counter. You might be able to get your 300mA down to 50mA.
The way i see it, designing the failsafe system to deliberately shut off is a real poor solution. It's like designing a parachute so that if you are a bit heavy on that day the 'chute breaks away so it doesn't get damaged.
(edit)After looking at your clock schematic, it seems you use the 7-seg display drivers as part of the clock itself, so you can't power them down to conserve energy, BUT you could drive the common cathode of all the displays with a transistor to ground, and operate the transistor base from the 7.35v junction in your backup supply. That way the displays will power down in a blackout but the clock count itself will be maintained, and current should be down to a handful of mA.