But the move from subjective indistinguishability to evidential indistinguishability seems to ignore an important point: meanings ain't just in the head. Even if two brains are in the exact same physical state, the contents of their representational states (beliefs, for example) can differ. The contents of these states depend not just on the brain state but also on the brain's environment and causal history.

You're assuming that there exists something like our universe, with at least one full human being like you having beliefs causally entwined with Obama existing. What if there is none, and there are only Boltzmann brains or something equivalent?

In a Boltzmann brain scenario, how can you even assume that the universe in which they appear is ruled by the same laws of physics as those we seemingly observe? After all, the observations and beliefs of a Boltzmann brain aren't necessarily causally linked to the universe that generated it.

You could well be a single "brain" lost in a universe whose laws make it impossible for something like our own Hubble volume to exist, where all your beliefs about physics, including beliefs about Boltzmann brains, is just part of the unique, particular beliefs of that one brain.