Best battery and adaptor for simultaneous arduino charge and power

Hello Arduino Forum!

I'm currently building a project that utilises 12V linear actuator, LED traffic light system, 4A dual DC motor shield, and arduino uno. The project is fairly simple. The 12V linear actuator runs in three cycles with the red LED indicating the first cycle, amber LED indicating the second cycle, and green LED indicating the third cycle. This will be run with a single arduino script.

Currently, I've been working on a circuit that uses 8 1.5V AA batteries in series. However, I'd like to be able to use a rechargeable 12V battery (with adaptor for mains plug-in) that employs a DP/ST switch so that the circuit is broken while the battery is charging from a mains supply but also can't be charged while the circuit is in use. Ideally this would allow the circuitry to be housed in a closed case so that it can't be tampered with or exposed to skin.

Any ideas are appreciated!?