The MM-100 is not wired like standard dual sided 3.5mm cables, so you'll need to ensure the balanced cable is made according to the MM-100 pin out. If you have existing balanced cables for other headphones it will likely not work.
The pin-out for MM-100 cable connector at the headphone side is the following:
TRRS:
Tip - L+
1st Ring - R+
2nd Ring - L-
Sleeve - R-
The wires in the headband are connecting via two TRRS connectors in parallel, so it's not important which side the cable is connected, left or right. Both versions can be used, single ended (TRS) or balanced (TRRS); the left speaker will always take only the left signal and the right speaker will always take the right signal.
Making this cable balanced will require splitting the L and R channels into appropriate pin-outs for your amplifier.
Here's a diagram for the internal wiring to give a better picture of what's happening: