I just ran into a Portable board doing very similar issues. A Couple years back, i had a portable that had the exact battery issue. I remember a broken trace to the 3rd cap, at the right and the Op-amp and resistor pack change out.
The LT1179 IC is getting noisy, and I noticed at a 6.2V input, the 5V regulator was a low 4.6V, and wouldnt go into proper regulation until the input voltage was near 7V. That aint no good.
So I got some LT1179 ICs on order, hopefully it works. Those little buggers are pricey!
Just above that IC is a highly proprietary resistor pack made for apple that forms the voltage dividers, and feedback networks for all those Op-Amps inside that IC. If one of those is bad, well.. your F****D, as I dont have this resistor pack mapped out. I need some fuming nitric acid and a vent hood to decap it and figure out its internal layout and resistor values.