2. It's called relativistic MOND. That exists as a category. RelMOND(for short) includes TeVeS, Einstein–Aether variants, BIMOND, and others.
3. I would wager that what we call the CMB is more of a local effect than a big bang remnant.
I lack the karma to downvote, don't blame me. :)
2. not a physicist but iiuc the relativistic MONDs have been disappointing? but not enough to rule out