'm unsure if I'm missing a major difference between these two servos which is causing the different behaviours, but I'm having trouble finding this if there is. Perhaps the amount of current being drawn for the larger motor is too high ?
Don't be offended , but what you are missing is a basic understanding of electronic and mechanics.
FACT-1
The large servo is designed to run of voltages between 6V and 7.4V .
FACT-2
It is a METAL-GEAR servo , meaning there are no plastic gears connected to feeble motors to limit the current. Even small plastic
gear servos can draw more than 1A in a stall (jammed).
The large servo would easily pull your 5V down even without stalling. (I would expect)
That servo needs it's own power supply an it would have to be a 6V walwart that can handle several amps.
You could get by with a 6V ,1A walwart until you put a load on the relay (connect the arm to something) but once you have a load you would need at least 2 or 3 amps.