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Hello Hiram,
There is no regulator internal to the device that can operate in the way that you laid out in your post. The regulators present on the device such as the output of Vref would be insufficient to run your system. The buck converter that is present is used explicitly to take the input voltage and regulate it to effectively charge the battery.
The simplest solution would be to take a two stage approach and attach a buck converter on the line to the system seen in the datasheet schematics. This would then buck the voltage to the system down to the desired 6V regardless of whether it is 12V coming from your adapter or some other voltage coming from the battery.