DC Battery Stability
I would use a 8.4V LiPo battery pack rather than 12.6V.
An 8.4 will would use a  DC-DC step-up boost converter.
A 12.8V would require a combination step-up/step-down converter.
Off the shelf solutions are few for a 1.2 A, 12V - 12V chargers and regulators.  

But you already have a 12V battery pack.  

You could use the versatile Linear Tech LT3652 Battery Charger  

Input voltages from 5V to 32V 
Will charge up to 2A
Works with Lead Acid, Li-ion, LiPo, and Li-phosphate.  





Example 8.4V Li-ion charger powered with 12V AC adapter: 







Example 12V Lead Acid charger powered with 5 VDC to 32 VDC : 









Then for the output regulation the Linear Tech LTC 3111 12V 1.5A Buck-Boost DC/DC Converter
With low battery shut off. 





If you want to switch between the AC adapter and battery without the voltage dropping to zero you can use a Linear Tech LTC4358 Ideal Diode Switch  . 





Or a Linear Tech LTC4412 Automatic Switch