My battery light has been coming on and off for a couple of weeks now and saying "Service Battery Charging System".

might be due to the battery being bad and may need replacement but if the battery is holding a charge then it might be due to a short out in some place

I drove Volvos for 17 years. Finally learned that a New battery had a short. Meanwhile, I'd changed out alternators. Switched it to a generator. It was a bad battery, from the get go. Just a thought.

Try jump starting, see if it starts then. Lights, radio, etc.. don't draw nearly the amps that your starter does. Batteries don't always give warning when then decide to "go south". If you can get it started by jump starting, stop by a local Autozone, Advance, etc.. auto parts store and have them do a charging system test and to load test your battery. They offer this service for FREE
Another Note: When was timing belt last replaced? If it's a 325 of that year it should be done every 60k (If it's 318 ignore the above belt issue)

The BATTERY Light coming on is telling the charging system is not charging the battery . Take it to a qualified repair shop . The abs traction lights coming on are due to low voltage .The charging system in your vehicle is computer controlled ,don't know what kind of mechanic you took it to ? Take it to a Chevy dealer

It could be the alternator not charging properly. Even though you're not having trouble starting the car, the alternator could be putting out just enough to handle charging the battery. But eventually it will go out. The battery light can mean either the battery is not charging correct from the alternator, or it's time to test the battery, which I think is good from your entry.

unhook your car battery for about a half hour. hook it back up and see if the lights are off now. all the codes should reset, drive the car over to autozone and have them test the charging system to see that it is operating correctly. also if your radio is of the anti-theft variety make sure you know the code to activate it and how to enter it.

Optomum Alt Charge is 14.5 so yours is Low And Junkyard Electrical Parts I NEVER Recomend I Would Replace Alt w/ Reg. NEW

The alternator test will likely indicate a charging problem from the alternator that is not keeping the battery charged up.
Regardless of the alternator test, it would be best to do an electrical draw test to determine whether there is an excess drain on the battery. This can be done with an ammeter between the positive terminal and the positive battery post. This test is done with key off, all accessories off, no lights on, etc. There will be a small draw on the battery in all newer vehicles, but it should be in the order of up to 50 milliamps. If you are getting more than 200 milliamps of current draw then there is a problem with an electrical system that will drain the battery over a period of days or weeks. Once you know this is true, the electrical systems can be methodically tested.

This is a common problem with these cars. I have had it come on also a couple of times. No need to worry unless it stays on.

sure sounds like a battery / alternator issue , take it to an autozone or checkers , let them test the battery and charging system as a courtesy , and a second opinion . thanks.