What we are trying to get across is that if the battery is in good condition and fully charged, the car should run off the battery alone (for a short time) unless there is a connection issue in the wiring between the battery and alternator.

The alternator will back feed the connection issue masking it until it can no longer do so. At that point it will either stop running altogether or caus a melted harness / fire.

Disconnect your alternator field terminal and see if the issue returns. It should as the alternator will no longer be charging.
IF so, run a 10G jumper from the Alternators battery post to the battery. this will prove what we are trying to tell you as the jumper will now back feed from the battery.