Tamiya ESCs can be confusing, they are different than most all others.

The specs aren’t calling for an external BEC, they are saying you must use a receiver with BEC. Meaning a receiver that can handle the main pack voltage, the ESC does not regulate the voltage down to traditional radio levels. I don’t know if the Futaba R603FF is capable of such.

When they say the BEC must be plugged into the receiver, they are referring to the 2 wire connector from the ESC, needs to be plugged into the receiver.

EDIT: To clarify, you can use an external BEC, just not required. If so don't plug in the 2 wire connector from the ESC, but be sure to insulate it/prevent any shorting.

Sounds like your issue is the lipo low voltage shut off is engaging. Either that or simply overloading with too much motor and/or too much gearing for the ESC.