Failover Management from AC to DC and DC to DC
In general, if your AC/DC converter puts out 13.5VDC@req'd amperage, all you would need to do is wire it in parallel with the battery.

That way, the AC/DC converter charges the battery &amp; runs the load when 'live,' and when the AC/DC converter isn't prodicing all of the load-required power, the battery will supply the rest. No logic circuitry required.