You cannot use a voltage divider for power.
Since you only need a small capacity, you could use a small number of batteries with a boost converter to reach the required voltage. Since these converters are relatively new, your highschool physics textbooks won't tell you about them.
You can buy battery holders for AA and AAA batteries in almost any configuration you can imagine. 8 in a 2x4 brick or long-and-skinny or anything.Many battery holders even include a switch.
Yes, you can just use 8 batteries in series. And yes, the voltage will slowly drop over time from 12V as the batteries convert their chemical energy into electrical energy. But because they are in series, they all will slowly drop.It's unlikely your circuit will studently stop working when the voltage drops below 12V. But at some point, it will fail, just like any other battery powered thing, even Voyager 1 which has plutonium battery.