Even though Loeb's theorem can't be derived by a translation of the standard proof with two epsilon large bounds, might there still be a different proof?

Again, Christiano et al supposedly prove the existence of a coherent distribution with certain properties. (Someone who knows game theory better than I do should work out what the proof depends on.) Any such distribution necessarily violates the probabilistic Loeb's theorem, for roughly the reason given in the OP's second-to-last section.

Is there an intuitive justification for the two epsilon large bound other than that it stops Loeb's theorem from being derived?

Not as far as this lay-reader can see. Again, the distribution must satisfy derivation principle #1. I can't tell if it must obey #3, though if it does that would seem to rule out a stronger version of #2.