Offhand, it sounds like either an issue with the cable connecting the battery to the motherboard (e.g. one or two data wires broken somehow but not the power/ground wires) or an issue with the battery controller itself. Unfortunately I'm not sure there's really a way to fix those other than opening the laptop, which it sounds like you don't want to do.
As for BIOS, I find that holding down F2 or Delete on boot is the standard on PCs but of course it's possible ASUS is using a different key for this.