Daniel Holz writes on a popular science blog Cosmic Variance:"The Copernican principle is a guiding foundation of cosmology. In short, it states that we are not in a privileged place in the Universe. A “random” observer will see the same Universe that we do." He further writes "Now suppose I told you I had a model which explained all of the observations, was based on general relativity, and appealed to no mysterious dark energy component... There’s one important caveat, however. The void model throws out the homogeneity and isotropy assumption. The Universe is now spherically symmetric, with a big hole in the middle. Even worse, we happen to be very, very close to the center of the hole."
Let's suspend discussion of what is more "plausible", dark energy or giant void, or what "plausible" even means as applied to the 'universe as a whole', or what justifications one might offer for the Copernican principle. There seems to be an overt non-sequitur in its application. Assuming that we are not in a privileged position does not imply that we are in a generic one. The third option, and seemingly the most "plausible" one, whatever that means, seems to be that our position is neither privileged nor generic. There are many like it, and there are many as well, that are nothing like it.
But much of Big Bang cosmology is based on extrapolating what we observe to the entire universe, in other words it is based not on the Copernican principle, but on its non-sequitur. The whole reasoning is uncomfortably reminiscent of Kant's antinomy of space and time, where one gets a contradiction by applying forms of experience to categories that go beyond any possible experience, such as 'universe as a whole'. Similar reasoning is also used when talking about the "wave function of the universe".
Is the philosophical status of Big Bang cosmology epistemologically different from that of most scientific theories in that it explicitly relies on metaphysical principles and reasoning? Is the argument from the Copernican principle salvageable at least as a hypothetical, or is it just a misapplication of concepts?