It wont be the battery, even if it was old/bad, the alternator itself will keep the car running (even with NO battery at all)...So problems is most likely alternator, or wiring (but dont think so). All i can say is "good luck", that water cooled alternator is not available just anywhere, and $1300+ from the dealer...
Hope it works out for you!
PS, the loose alternator belt idea is wrong, same belt runs the water pump and power steering...if either one was not working she would have known it!!! (eg; NO steering, and overheating!)