Stepper motor properties versus battery life

I assume that it's not regardless of size?

Torque depends on permanent magnet strength as well as winding current, so it is likely that a larger motor can deliver more torque for the same current.

I have to admit that I have no experience in geared motors or servos at all.

You should get some experience before making the decision. Gearmotors are much better for battery operation.

Pololu has a good selection of gearmotors, but since you have not told us what your torque and power requirements are, it is impossible to offer specific recommendations.