Since you are willing to suppose that these initial conditions are different in each trial, why not analogously suppose that in each trial of the digit of pi version of the experiment, that you compute a different digit of pi.

Yes, I think that would work - if I remember right, zeroes and ones are equally likely in pi's binary expansion, so it would successfully mimic flipping a coin with random initial conditions. (ETA: this is interesting. Apparently pi's not yet been shown to have this property. Still, it's plausible.)

or, more generally, that in each trial you compute a different logical fact that you were initially completely ignorant about.?

This would also work, so long as your bag of facts is equally distributed between true facts and false facts.