I found some schematics which actually make use of the diodes in the 6AV6 by Googling "vintage radio schematic 6AV6" and looking in "Images".
I looked at a few of those with the 6av6-12av6 type. It looks like most of those are using the diodes as detector, their design purpose, to strip the AM carrier from the audio signal.
The amplifier usage I've seen attempted so far seems to be as a clipping diode, and that might be all I could end up with. My thought would be to adjust the cathode resistor so that the ac peaks of audio signal could pull the cathode voltage upwards smoothly, in a more linear way than clipping, and result in a smooth compression. The specs say that a 10vdc on the diode plates can draw 2ma each, so 4ma could lift cathode voltage slightly given the right Rk. The problem I see is that it would only effect the positive swing of the ac signal, and result in an imbalance possibly. On the other hand, if the upswing were lowered, that should lower the total peak to peak voltage.
The other issue is, once the Rk is sufficiently "loose" for a <4ma current to change the bias level, the triode may be incapable of much, if any amplification. Unity would be okay, since the effect is what I'm after, but it may not even be capable of unity. I wish I could sim the whole thing, but I have no sim software, and have yet to learn how to configure one if I had it.