>Also, can't we explain the dipolar variation with some motion in some direction we haven't accounted for (in our local cluster for instance)? I imagine that quadrupole and octopole variations are harder to explain through motion.

As far as I know that shouldn't be the case. There's no standard frame of reference with respect to which you can move, so for nearly all intents and purposes the earth might as well be standing still. There's also not really any way to 'move' with respect to the CMB; it's always the same time (i.e. distance) away. And at the time of it's creation particles were (presumably) moving in random directions at random speed, with no specific average (or if you want to be precise, with a Lorentz invariant speed distribution).

Searching for 'CMB Lorentz violation' does turn up a few results, but most of those seem to be pure conjecture, I couldn't find any experimental evidence either way.