I mean, clearly it isn't in this case. But given that the digits of 2^n are cyclical at each decimal position, it does feel like this should fall out of some sort of chinese remainder theorem manipulation.
True. It might also just be that the question hasn't attracted the attention of number theorists, and finding a proof wouldn't be unreasonably difficult to an expert in the field.
Nope, it's not that easy in this case. E.g., Erdős conjectured in 1979 that every power of 2 greater than 256 has a digit '2' in its ternary expansion [0]. This makes sense heuristically, but no methods since then have come close to proving it.
Digits of numbers are a wild beast, and they're tough to pin down for a specific sequence. At best, we get statistical results like "almost all sequences of this form have this property", without actually being able to prove it for any one of them. (Except sometimes for artificially-constructed examples and counterexamples, or special classes like Pisot numbers.)