Proper way of designing a battery backup system with 12V/5V dual output AC power supply?
You always start with specifications not a design idea.


V , I, Load  impedance  ESR of source 
Then put in min/max range or tolerances.
Then add efficiency and budget.
Then compute power delivery and losses allowed.
Then add back up time and Watt-hour capacity needed


then choose Voltage of battery from Ah ratings needed and cost.



Then start the design.