I don’t understand your argument for why #1 is impossible. Consider a universe that’ll undergo heat death in a billion steps. Consider the agent that implements “Take an action if PA+<steps remaining> can prove that it is good.” using some provability checker algorithm that takes some steps to run. If there is some faster provability checker algorithm, it’s provable that it’ll do better using that one, so it switches when it finds that proof.
I don’t understand your argument for why #1 is impossible. Consider a universe that’ll undergo heat death in a billion steps. Consider the agent that implements “Take an action if PA+<steps remaining> can prove that it is good.” using some provability checker algorithm that takes some steps to run. If there is some faster provability checker algorithm, it’s provable that it’ll do better using that one, so it switches when it finds that proof.