If you have all the info, AND the given battery has a high enough voltage, why bother with an inverter to bump it up to a higher AC voltage just to knock it back down again and make it DC?
A switching power supply takes the incoming AC voltage, rectifies and filters it to a DC voltage. The voltage regulation is done then by switching on a transistor (simplest form) for x-amount of time, and then switched off for y-amount of time. This is done at a high frequency and on only long enough to charge and hold the output at the specified voltage using caps to hold the charge and help deliver some current.
A modern PC power supply takes line voltage, converts it to DC and then runs it through the switching circuitry to drop it down to 3.3V, 5V, 12V, and -5V and -12V (These last two are the least used and have very low current out) They do this with better efficiency than they did years ago. So, the trick is to design your switching power supply that runs off the battery as efficiently as possible to minimize losses. (Now days, a good PSU in a PC will run at about 80% efficiency, or better under most loads. Cheap power supplies? Well... think a 5v ceramic pumped up on 120 or 240, when they get severely overloaded. Most of the time before delivering their rated power out.)