Sorry to join this discussion late. Indeed we have no tube mics in production but I have studied them extensively, built more than a few and thought several times about doing a tube model. Also, this question is certainly not unique to tube mics. Our standard FET first stage is precisely controlled fixed bias at the gate with the drain driven by a current source, so you can say that I'm a fan of "fixed bias."
But as Oliver mentioned, there aren't any standards for what we call things in microphone electronics, so describing the differences we're talking about as "fixed" versus "self" bias is a bit of a misnomer. Things are different than with voltage amplifier stages; the relation between the first active stage and the capsule is quite complex as the impedance the capsule sees changes according to the current drawn by the grid or gate of the first stage. This is in addition to anything that would be caused e. g. by a cathode bypass capacitor.
My design approach to date has been to get rid of as much "sound" as possible from the electronics. (Thank you Paul Klipsch, good sound is the absence of bad sound.) If I can prove to myself that the signal voltage developed across the polarize resistor is unaffected by the first amplifier stage, I'm happy (and that's hard, when it drives you to operate the input impedance in the gigohms). But, that is by no means the only possible approach. Circuits whose characteristics shift with signal level (I call this signal-history distortion) make a certain sound and this might be entirely desirable in some instances. Many historically valued mics exhibit this characteristic. I try to focus on what aspect of the transfer characteristic is judged to be advantageous (who says you can't build a magic limiter/compressor into the mic?) rather than trying to associate it with a given circuit topology. I would recommend looking at what the circuit is doing, rather than trying to hang it on presence or absence of some resistance and impedance in series with the cathode. In this case I think the real question is whether and how the grid voltage changes with signal level.