Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empirically privileged reference frame?
Your mistake is to assume that the cosmic microwave background constitutes a universal rest frame, because it doesn't.
In an FLRW universe there is a frame called the comoving frame or proper frame that is particularly mathematically convenient. This is the frame in which the comoving distance between all inhabitants of that frame is constant, so all the "stuff" in the universe is mutually stationary (the comoving frame factors out the Hubble expansion). Given that we expect all the "stuff" in the universe to be created in a similar way we would expect it to be approximately stationary (in a comoving sense) so the sum total of everything, matter and energy, acts as a reference point for the comoving frame.
So there's nothing special about the CMB. If you ignored the CMB and measured the Earth's velocity to all the galaxies we can see then you'd expect to get the same result as measuring the Earth's velocity relative to the CMB. The CMB occupies the same frame as everything else because it was created in basically the same way. The only special thing about the CMB is that gravitational interactions haven't given it various peculiar velocities as has happened for large aggregations of matter.
The universe not being isotropic is a perfectly valid observation, as far as relativity is concerned.
However, doesn't this give this "universal rest frame" and its associated time an empirically privileged status, even though physical laws work just fine in every reference frame?
Yes, it does, but that's not a problem. Relativity doesn't prohibit a physical system from selecting certain reference frames that are "preferred" in the sense that they're more convenient to work with. In fact, this doesn't happen only with the universe. Even for a small system, like two particles, there is one empirically privileged reference frame, the center-of-energy frame in which the total momentum is zero. (This is a spontaneous breaking of Lorentz symmetry, if you appreciate such terms.)
But the fact remains that the fundamental laws of physics are perfectly valid whether you're in that empirically privileged frame or not. The only sense in which it's empirically privileged is that it's convenient, but it's not a necessary choice.