My money is on a bad fuseable link (or equal) between the alternator and the battery.

IMHO, the fuseable link wire probably went out when the alternator failed. The motor will run (for a while) off the battery without the charging circuit working, but turn on the head lights and it is too large of a load without the charging circuit, which kills the battery. If the head light circuit was a dead short it would blow a fuse, stop the motor when the lights were turned on, or worse start a fire in the wiring.

A fuseable link would explain two replacement alternators and a new battery not solving the problem. Just my 2 cents worth.

.