If the battery light is illuminating intermittently, your problem is somewhere in the charging circuitry. Starting logically, have your mechanic do a battery stress test. This is something for which his facility is equipped. The test puts stress on your battery and shows whether it is in good shape. Next, take a look at the belts just to make sure they are tight and in good shape. It may seem as if this is an obvious check but it isn’t. There should be no shiny spots or wear marks.
Next, have your mechanic perform load testing on the alternator to make sure that it is working correctly. It could easily be that a secondary winding is starting to fail and that may be causing the issues. Equally as important, though, is the output circuitry. Be certain that you have a diagnostic routine performed on all parts of the charging system, including the various circuit boards. All it takes is one flat pack failure or intermittency and you can have the issues you are talking about.
Be sure that you have all parts of the electrical system run or tested to make sure that there are no shorts or opens in the wiring. Yes, this can be expensive, but it has to be one.
I noticed you also indicated that your transmission is having issues. Since your battery and charging system are showing problems, it is possible that one or more of your transmissions sensor arrays or solenoids could be failing. Watch for any error codes that might indicate transmission problems. Have the solenoids tested independently (they should open and close with a sharp click when energized) just to be sure they are working correctly. A sticking solenoid can cause stumbling and performance as you described.
There are so many possibilities. Have your mechanic perform a general diagnostic to see if there are any other areas of trouble that might be identified.