Would a battery be considered an independent or dependent voltage source?
there are many different battery models some can be as simple as a voltage source with series resistance some are more complicated.

You're correct in thinking that there is a dependent source in more complicated battery models the lifetime is dependent on the current that has already been discharge through the battery. Such as shown in the circuit below:

.
Source:
https://www.edaboard.com/showthread.php?214394-how-to-make-a-battery-model-of-NiMH-on-HSPICE  

The model uses in RC circuit to model the capacity of the battery and the current withdrawn from the circuit.