This post does solve the class of problems Eliezer was talking about (I call it the "fair" class), but my previous Löbian post used a different algorithm that didn't do utility maximization. That algorithm explicitly said that it cooperates if it can find any proof that agent1()==agent2(), without comparing utilities or trying to prove other statements, and that explicitness made the proof work in the first place.

Your proof fails here:

Therefore, under the supposition, agent1 will choose "C"; similarly, agent2 will choose "cooperate".

It's extremely difficult for the agent to conclude that it will make any particular choice, because any proof of that (starting from your assumptions or any other assumptions) must also prove that the agent won't stumble on any other proofs that lead to yet higher outcomes. This amounts to the agent assuming the consistency of its own proof checker, which is a no go.

As far as I can see right now, making the payoffs slightly non-symmetric breaks everything. If you think an individual utility-maximizing algorithm can work in general game-theoretic situations, you have a unique canonical solution for all equilibrium selection and bargaining problems with non-transferable utility. Judging from the literature, finding such a unique canonical solution is extremely unlikely. (The transferable utility case is easy - it is solved by the Shapley value and my Freaky Fairness algorithm.)