Battery light came on my 2004 saturn vue. We replaced the battery because it is six years old. light is still on and no air conditioning

2004 Saturn Vue manual transmission. Now at 150K miles and I just replaced the right wheel bearing, serpentine belt and battery last month. The odometer light burned out. Then suddenly, last night, my battery light intermittently comes on. Took her to Autozone and altenator/battery both checked out fine. We were 60 miles away from home so we decided to cut our trip short and turn around. We made it 45 miles and the entire dashboard lit up (ABS/traction/service light). The battery died and we had to have her towed 15 miles home. Now our Vue obviously won't start because the battery is completely drained. I'd prefer not to spend another $1,000 figuring out what's wrong after just dumping $700 on the other repairs. I've come down to either the BCM... a bad ground wire to battery... or possibly a bad altenator (last resort). Just extra info: There's a burning scent (like burnt cheerios) after an extended time of driving. What was your solution?

Also, check for corrosion on the negative battery cable and terminal. If bad replace it. Also, you can test the alternator by diconnecting the ground cable temporarily while running and f the ar shuts down it means the regulatr is not working properly any more.

WHEN THE BATTERY LITE COMES ON THE DASH IT IS TELLING YOU THE VOLTAGE FROM THE ALTERNATOR IS WORKING, TO CHECK THE ALTERNATOR WITH A VOLT METER ON DC WITH THE ENGINE RUNNING YOU SHOULD GET A READING OF 13.5 TO 14.5 VOLT WHEN YOU TEST THE BATTERY TERMINALS IF THE VOLTAGE IS LOWER THEN 13.5 VO0LTS YOU ALTERNATOR IS BAD