I agree that Godel machines probably don't work for similar reasons, though didn't notice your post until now. Unfortunately, I also think that your non-self-referential alternative runs into similar issues (where subsequent agents use successively weaker axiom systems, if they follow the same general design). I have been thinking along these lines independently, and I think resolving either problem is going to involve dealing with more fundamental issues (e.g., agents should not believe themselves to be well-calibrated). I've queued a rather long series of LW posts establishing what I consider the current state of affairs on FAI-related open problems, a few of which concern this issue (and of which the OP is the first).

I wrote a comment explaining why Schmidhuber's Gödel machines probably don't work either

I'm not sure about that billing - the comment seems to give up before much of a conclusion gets drawn.

Gödel machines are like GAs. The problem is not so much that they don't work at all - but rather that they may work too slowly.