Most modern op-amps don't respond to class A biasing. One of the reasons is because they have diamond buffer outputs (AD845, AD811, some others). All you can achieve is a slightly different and probably worse sound. The only op-amps I found responding to class A biasing were CA3140E (it's a joke to use it for audio) and AD825 which sounds different with no class A bias (I would say best), another way when biased to sink into class A, and the other way when sourcing from class A bias.
Using compensation pin as output is a typical forum hoax. It's sounds terribly thin and bright, much worse than normal output of the AD744. If it improves overall sound then it means you have real problems with your rig's sound signature being far from neutrual/transparent. There might be many reasons like: bad source, bad amplifier, high output impedance (OTL tubes, resistors in the path) driving low impedance phones, poor quality interconnects, etc.
Summarizing:
3.4mA class A biasing for AD825 might be interesting... same as this chip alone.