There's the same voltage between the positive and negative terminals on a cartridge whether it is wired balanced or unbalanced - there's no difference between the two.
Also, a step-up transformer can be wired as balanced input with unbalanced output, or balanced at input and balanced output, or unbalanced input with balanced output, or unbalanced at both input and output. There's a multitude of options. The most common is unbalanced input and unbalanced output. In all cases the voltages are the same.
Yes, Rega's wiring shouldn't be a problem. I'm still trying to work out exactly what is wrong in the OP's setup but I'm confident it is that Rega's wiring uses just one signal negative as the ground for the arm but the SUT is linking both signal negatives together. Or maybe it isn't It's a lot easier having the equipment in front of you than diagnosing problems via forum posts.