Ah the persuit of better sq.
A few things considering your experiment:
(1) You're not concerned about voiding warranties in your experiment.
(2) Yo don't mind if one or more components aren't functioning after your experiment
Having tossed those two considerations aside, here is what I think would work best without being a true full Balanced LO to Balanced LI connection;
Connect each of your HP channel this way:
TRRS. XLR(pin)
Left +. --->. 2
Left - --->. 3
Left - --->. 1
Repeat and rinse for the right channel. Although not the best configuration compared to a true balanced line out, it should provide you with roughly 20 to 30 dB of common mode rejection. You use a 2 wire cable with a ground shield (3-wire connection for a balanced channel).
Now my honest opinion about this: unless you run miles of cable and/or you're in a electrically high noise environment, you're results won't be that much better. I believe people's perception of a better sq is more related to higher voltage levels afforded with a true balanced connection, that's just my opinion.