You are right in that the alternator should do its job after but ive seen where power draw will spike and thats were the battery comes in and help provide the juice. In my case i have currently 200K with a rebuilt transmission that needs to have the sonnex sleeves installed in the valve body any way in 200K alternator was replaced 1 time and im on my 3rd battery first 2 the dealer paid for. but only time i had trans fail safe was when the alternator was not able to provide the juice...

my cousin has a ford f550 and he had issues with limp mode i went and tested batteries etc and they were sitting with the truck off at 12.7 and 12.5 not bad but doing the alternator test i noticed times it was charging 14v and times it was sitting at 12.9 and spike to 13.5. Told him to take it to the dealer and have then verify my findings and i was right alternator was going out and was putting the truck in limp mode which would go away at times but in got to the point where it was happening and at that time it wouldnt go away.

so in your case it might just be the alternator thats going out.....you can try buying a regulator for the alternator idk the cost on the reman but ive always kept my alternators on my other cars and just replaced the regulator on the back which is where the brushes are and the part that actually wears away *so does the shaft but not as fast as the brushes.