No, a failed Smart array battery won't bug your box, but any one or combination of: a failing raid controller, especially the onboard ones, a bad motherboard, bad power supplies, bad power AC/DC power regulators, will.
Yes, are are seeing a sensor for the battery, but that is just coincidence for all the other stuff that can go bad over time for a decade old box.
All the ProLiant sensors just tell you if something is present or not-present, and doesn't account for "works some of the time".
You can try a new battery swap, but the cost of the battery will be the same as replacing the entire box, your call.
You can also try to drop in a better smart array card, but the downtime to replace the entire box is just as long.