OK so you have eliminate the transmitter and receiver.
Have you checked the individual cell voltages under load?
Also check that the battery type on the ESC is selected correctly, if you have say set it for LiFe cells it will miscount the number of cells and apply an incorrect low voltage cut off.
I am pretty certain the problem lies in the battery, especially if you have tried another ESC, either faulty cell or possibly a bad solder connection to the connector, if the connector get warm when the motor is running then it's a sure sign of a high resistance in the connector generating heat and reducing the voltage seen by the ESC.
BTW something like the MPX Lipo saver ( especially the ammeter version) or SM Modelbau Unisens E sensors are a good way to detect battery problems.