Your battery sounds like it is either not high enough mAh (affects current) or just way too old.
I know LiPo setups demand high current batteries that come with high mAh ratings. How old are the batteries, and how's the mAh? If it's running for a few minutes then stopping I believe that the mAh is just way too low, or your charger has some issues.
The Built in BEC will cut off the voltage after the voltage drops below a rating.
I am a man of few words.