Thanks for taking the time to try puzzling this out, but I suspect it's just interestingly wrong. The magic seems to be happening in this paragraph:

Joe prefers Option 1. Therefore he anticipates that he will choose Option 1. Therefore, his current utility is 2U(1/2). But what if he anticipated that he would choose Option 2? Then his current utility would be 2U(1/2+p/2). So he wishes his k were smaller than U-inverse(k), meaning he wishes his U(x) were closer to xU(1). If he were to modify his utility function such that U'(x) = xU(1) for all x, the new Joe would not regret this decision since it strictly increases his expected utility under the new function.

I don't see where U(1/2+p/2) comes from; should that be U(1)+U(p)? I'm also not sure it's possible for the agent to anticipate choosing option 2, given the information it has. Finally, what does it matter whether a change increases expected utility under the new function? It's only utility under the old function that matters - changing utility function to almost anything maximizes the new function, including degenerate utility functions like number of paperclips.