DRV135: maximum audio transfer distance

This should be possible. The only concern I would have would be with using coaxial cable for a differential signal. In this case, the shield of your coax would be driven to a voltage, not to ground, which could cause issues if the cable shield is inadvertently shorted to ground. Additionally, you would likely have worse noise immunity than using a shielded cable pair (such as the Belden 8451 recommended in the DRV135 datasheet). Stability of the DRV135 may also be affected by the cable capacitance of such a long coaxial run as well, requiring some isolation resistance between the DRV135 output and the cable input.

With the coaxial cable, the two signals (+ and -) will not be balanced, and will have different amounts of noise coupling - in other words, the noise will couple differentially, rather than as a common mode signal. At the receiving end, the INA137 will reject much of the common mode noise (up to its CMRR), but any differential mode noise will still be received along with the signal.

With a shielded twisted pair, the two signals will be more balanced - noise should couple relatively symmetrically into both signals, becoming a common mode signal which then gets attenuated by the INA137's CMRR.

That being said, if you want to use coaxial cable as your interconnect, as long as your source and receiver do not share another common ground connection (such as through a building ground), simply using the coaxial connection as a single-ended cable may still provide good performance. In that case, you could use an audio op amp (such as the OPA1678) - again with an isolation resistor between its output and the coax for stability - to drive a single-ended signal into the coax.