According to Jameco:
Servos are controlled by sending an electrical pulse of variable
width, or pulse width modulation (PWM), through the control wire.
There is a minimum pulse, a maximum pulse, and a repetition rate.
I'm ignorant of the effect of a voltage regulator on pulse width modulation, but based on your experience, I suggest that it is detrimental.
You can confirm this if you have access to an oscilloscope with which to view the pulses directly from the circuit and then again from the output of the regulator. Correction, view the input to the servo without the regulator and then with the regulator.
According to Roger's Hobby Center, a 2S lipo battery, nominally 7.4v will reach voltages as high as 8.4 volts. This moves the concern of 6v deeper into the question.
Another website, pololu.com references "old style" servos as being designed for 6 volts in the form of 4 batteries at 1.5v, while manufacturers are embracing the newer battery chemistry and producing servos which will operate at levels as high as 9 volts.
Additional research (the data sheets for this model servo) show that it is designed to take 4.8 to 7.2 volts. The full charge voltage of 8.4 will cause the servo to run a bit faster, but I don't believe there will be long term effects, as the voltage will drop into the nominal range in short order.
Another piece of information I gathered somewhat indicates that a regulator will (or should) work, but it has to have the current capacity to handle the servo. I missed that figure in my research. This led to me discovering a caution that your circuits should have a common ground. If you don't have a common ground, your answer may lie there.