Does the ESC have a low voltage cutoff? If so, the LVC could be set too high, so that when the battery voltage sags under load (and it will sag), it drops below the LVC. At that point, the ESC cuts power to the motor, as it is designed to do.

The key is to first determine the LVC of the ESC. Ideally, you would then want to measure the battery voltage under load to see if is drops below the LVC.

- Jeff