Its highly unlikely that the size of the battery would be causing probs. But maybe the battery is old??? Then it would need replacing because of its age and lying around uncharged....
If the alternator has been serviced and checked by a competent shop then also have the cutout checked/replaced. The alternators had built in cutouts in most models, some of these after going bad have been sidelined and thier duty given to aftermarket cutouts.
Lets see.....oh yes the fan belt should be in good condition and not loose. There should be no high current drawing items like halogen lamps etc and the wiring should be good.
Have I covered all causes of electical mysteries??? Uuumm needs thinking....