I'd be inclined to borrow a known good/young battery from a pal and try that. Four years isn't that old, but it depends how the car's used as to how long a battery will last.
I heard somewhere that one failure mode (arguably designed in) of car batts is that bits of lead fall off the electrodes and collect at the bottom of the housing, eventually shorting out plates. If this was just starting to happen in yours, it could explain what you're seeing, I think :-/. Once you drive off, maybe it shakes the loose stuff around a bit and breaks the short? Can't really see why it would pulsate like you're seeing TBH though.
Or maybe I can? If the voltage regulator is seeing a voltage that sometimes suggests charging is required and sometimes not, maybe it can get into some sort of cycle with hysteresis where it's switching the brush current on and off intermittently?
Excuse my vague, meandering thoughts please!