See the battery doesnt matter. The BEC is the one thats modulating the current and voltage to the servo. You could have a battery capable of providing hundreds of amps at high voltage, but the BEC is the weakest link in the circuit. Heres what you want to do. Measure the voltage across the black and white wires going to the receiver, and then vary the throttle. The baseline voltage should be around 6 volts, see if it drops a bunch when you punch the throttle. If thats the case that means somethings wonky with the BEC or some other part of the ESC.

If that doesnt happen, then do the same thing on the wires powering the servo, that'll tell you if the receiver is somehow to blame.

Out of curiosity, what receiver and servo are you using?