are there theories in cosmology which do not incorporate/use the cosmological principle?
There are some inhomogeneous and anisotropic cosmological models that were analyzed and studied in past, like the Lemaitre-Tolman-Bondi dust, the Oppenheimer-Snyder collapsing ball, etc. But in the meanwhile the empirical support of the cosmological principle is strong and there are lots of phenomena that are considered proofs of it, e.g. large scale distribution of matter (galaxies, quasars), CMB, X-ray background, peculiar velocities, etc. IMO the only way to take some of these models seriously is to assume that they are equivalent to the standard cosmological model at observable scales, for example like this one. This would shift a little bit your question to whether there are currently cosmological models in which there is assumed inhomogeneity or anisotropy at very large scales. The answer to this is yes, even within the standard cosmological model the cosmological principle is not expected to apply at arbitrary large scales because of the finite size of our inflationary bubble.
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