That "battery power" is a mysterious thing and I think we're looking at a colloquialism that doesn't really tell us the whole story.
In "Mudd's Women" the Enterprise
also had to rely on "battery power" after it had used up its dilithium crystals.
The Making of Star Trek
is brutally clear on the issue: You have matter-antimatter energy for the "star-drive", (reactor) fuel for the impulse drive (later established to be nuclear fusion) and - battery power.
I believe "battery power" here is the equivalent of a nuclear fission reactor. After all, these "atomic matter piles" mentioned in "Court-Martial" have to mean something.