Truth be told, I don't have much faith in the battery tests that AutoZone and the like do anymore. Just because I have had many problems with batteries in the past that have tested "Good" ... they have sent me on huge wild-goose chases as a result. The second I replaced the battery anyway? The problem went away. The net of it is, have you tried swapping out the battery just to see if the problem still persists?
Part of me wouldn't be surprised at all if you replaced your seemingly "Good" battery, if the light went out and the problem disappeared.
PS- how old IS the battery...?
2000 Chrysler 300M [Daily Driver] 158,000 miles as of Nov. 2014
1996 Chrysler Town & Country LX [Always and forever the favorite.]
2000 Chrysler Town & Country Limited [Gone, but not forgotten.]