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You do it at the Planck temperature (1.4e32 K), obviously, giving one Planck mass (22 micrograms) per nat, or 15 ug/b. This probably has some physical relevance, but I'm not sure what.

Isn't that more to flip a bit than representing a bit, though?

Another angle might be to look at the Bekenstein bound, i.e. that if you have a physical system of mass-energy m, there's a maximal number of bits B that it can contain. Though that also would need a system-containing volume (surface area?) which would also be arbitrary.

You also have to convert mass to weight, and for that purpose you need to choose a gravity, which is also arbitrary.