Indeed, rather than trying to go with a 12-24 VDC transformer system, two 12V batteries seems the most logical solution.
Depending on how fancy you want to be:
Hook into series to power unit. Charge individually
Switch to parallel when charging, so you're charging them at the same time.
3. Really fancy: Use a few diodes so the batteries are in series when providing power, in parallel when being charged. I think 3 would be sufficient. Note: This is using different terminals for charging and supplying power. With 5 you should be able to use the same terminals for 'everything'.