Not a quantum physicists, but IMO Bekenstein bound is not applicable here, because quantum laws are non-deterministic, therefore you can describe the structure of a system, but you cannot describe how it will evolve. Quantum randomness might be in the very essence of how the brain and mind works.

Quantum randomness being necessary hardly seems like it would have profound practical implications since augmenting a digital computer with a geiger counter would be trivial.

It's much easier than that. Hardware random number generators are often based on something like a reverse-biased diode. Electrons migrate across and it's entirely random when it happens. Amplify and count them and you get a great source of entropy.