RE science confirms that the "CBR" is in fact "moving with us", whatever that is supposed to mean: it has been observed to be isotropic. This means that its spectrum is the same in all directions; there is no observed blue- or redshift in any part of the sky (to my knowledge; this has been disputed but I have not seen any references).
Um, yeah. That part in bold means we aren't accelerating through the universe. It makes no sense to state that the CMB is "accelerating" with us. How exactly can background radiation "accelerate"? Are you suggesting that the photons created at the surface of last scattering which are coming directly "toward" us somehow "slow down" at a rate of 1g? That's nonsensical. And considering that you invoke relativity in your defense of the FE "gravity", I'd suggest you not ignore the invariance of the speed of light.
We should see blueshift. A hell of a lot of it. We wouldn't expect to see it until we approach more significant fractions of the speed of light. That's why we don't in the real world. But at constant 1g we'd be at extremely high velocity very, very quickly. For example we'd reach the moon in about 2.5 hours and the sun in about 2 days starting from rest at 1g acceleration.
What is the force which supposedly accelerates the earth anyway? How exactly does it manage to accelerate us (and everything) at a
constant 1g? The faster we go, the more "oomph" is required to maintain that level of acceleration.
How does that work exactly?
Conventional theories of gravity, though they change, explain the universe pretty damn well. The FE explanations tackle one bit at a time without any regard to the larger consequences. The explanation of why we feel gravity (the constant acceleration of the earth through space) could explain the isolated observation of gravity on earth from the perspective of one person.
But what about planet and solar system formation? What about black holes, how do they form? What causes supernovae? What about stellar evolution? FE theory falls silent because it's not created to attempt to explain the real word; it's created to attempt to explain one singular, isolated question.