To give a boring answer:

If we are assuming there wouldn't be any other major changes to the political structure (e.g. no bayesian party in congress) then the effect on policy outcomes would be fairly minor. For better or worse the president doesn't have that much direct power, and has to work with a lot of other interested groups.

Also I think people underestimate the domain specific knowledge in politics, there's no reason to believe that being rational would make Eleizer a particularly effective politician any more than a good doctor or lawyer.

The main specific power the president has is in publicity, so Eleizer could probably increase attention on existential risk and FAI issues, but how much concrete change that would make I don't know.