OK, as I mentioned before:
1) Microphone cartridges attach to a transformer before leaving the mic. That's why they are balanced. The floating ground comes from the center tap. If you just attached the cartridge up to the XLR it wouldn't work properly.
2) Microphones don't have a volume pot attached after the cartridge, or the transformer. It wont work.
3) Your amplifier does not have a balanced input. If you don't have one side of the pickup grounded, it wont work.
If you do want to do it, the easiest way is the way EMG does it. Take an op amp. One side of the pickup goes to (+), the other end goes to (-), and the connection between the two coils is the ground. Since humbuckers are differential in nature, that would give you the plus and minus swing.
And you probably wouldn't hear a difference. There's very little reason to balance the signal right from the pickup.
Humbuckers are already differential, that's how they cancel hum. Mics are balanced because they have VERY little signal. The coils are about 10 Ohms. The internal transformer steps that up and makes it balanced. This is to kill hum induced in the mic cable, not the mic. It has little to do with the signal voltage swing.
And still they need a preamp at the end. The coil doesn't pick up much noise partly because it's low impedance.
As far as:
The q-tuners are super quiet, so I can't really attest to an A/B comparison where the balanced signal was quieter than the unbalanced. The gain structure and impedance are so different that any comparison was not very practical. I will say though that I got a very hot signal that was very clean and VERY quiet!!
Q-Tuners are just humbuckers. They aren't even shielded. The sideways coils help a little, and they have very little ungrounded metal in them. There is nothing in the design that makes them quiet.
The real test is to sit in front of your computer monitor... do you hear a buzzing noise? That's the lack of shielding. But shielding wouldn't look good in a clear pickup.
The gain structure and impedance of Q-Tuners is not all that radical. A BL-4 is either 12k or 14k. That's actually on the high impedance end of passive pickups.
The reason for this is they use air coils that don't have metal cores, so the inductance is lower. That's what helps them have the extended top end. But the lower inductance also means the output would be lower than a pickups with metal cores in the coils, so they have to wind them hotter. But you can wind pickups hotter with neo magnets without them getting muddy, especially with low inductance coils.
The neo magnets also make up for the low inductance, and don't contribute any inductance of their own. You will notice that the pole pieces don't touch the neos. This prevents the top end from being over the top by knocking it down a notch.
The other aspect of the Q-Tuner is the "side winder" setup of the coils. This gives them more of a single coil tone. The old Bill lawrence L-250 Strat sized pickups are made like that, but they only have the single blade pole in the middle.
So Q-Tuners are a very nice design, I like them, but nowhere as radical as an Alumitone or Transsensor. They don't even have coils, except for the small built in step-up transformer.
Obviously I have some experience with neodymium magnets and coil arrangements.

And you can see everything going on inside a Q-Tuner. I wont show what's in mine though, but you can sit in front of a computer monitor with no buzz. Shielding isn't pretty however.