What frame of reference in the universe is (most) rotation-neutral? I've read on Newton's Bucket and the pseudo-special frame of reference bound to CMB but I didn't find anything that would bind the two well - so, is the CMB frame "rest frame" for a spinning body not locally gravitationally bound? If not, is any other?

Let's perform a thought experiment: we have a space probe with a powered Control Moment Gyroscope which allows us to put the probe in arbitrary spin, and accelerometers that measure local acceleration in every direction (without care what's the source of that acceleration: gravity, engines, centripetal force etc) placed at extended locations on the probe (way off center of mass).
We place the probe squat in the middle of the Giant Void so it gets as little local disturbances as possible from nearby celestial bodies (since there are none; all very distant.) Then we drive the CMG in such a way as to minimize readouts of all the accelerometers.
Will we reach a flat zero readout? And with the readout so minimized that any operation only increases it, will be the frame of reference bound to the probe non-rotating relative to the CMB-based pseudo-master frame, or will it be something else - bound to what (besides the probe)?
 A: Yes, there exists a frame in which all accelerometers can in principle reach zero.  If fact, there are infinitely many such frames.  In some of those frames, the CMB would appear isotropic; in others, it wouldn't.  (I'm assuming that you're neglecting the radiation pressure of the CMB on the probe, since that can be made arbitrarily small by making the probe small enough.)
The statement that accelerometers register no acceleration in an inertial frame (i.e. a frame in which constant-spatial-coordinate curves obey the geodesic equation), which follows from the equivalence principle, is a statement about the fundamental nature of spacetime - it would hold true in any universe described by general relativity, regardless of that universe's initial or boundary conditions, and in particular it has nothing at all to do with the existence of the CMB.
The fact that there exists a frame in which the CMB appears isotropic is a statement about the initial conditions of our universe - it certain doesn't follow from general relativity.  And the existence of the CMB does not in any way affect the readings of locally moving accelerometers (except through a truly tiny amount of radiation pressure, as mentioned above).  Put another way, general relativity just says that the universe appears locally like Minkowski spacetime in small enough regions.  The existence of a CMB frame asserts that there exists a global frame in which the universe is (approximately) space-translationally and rotationally invariant (though not time-translationally- or boost-invariant) - a completely logically independent statement.
Also, note that "rotationally invariant" does not mean "looks the same no matter how you're rotating" - it means "looks the same both before and after you begin and end rotating, so that you are no longer rotating but are now pointing in a different direction from before."
