cosmological horizon - How can there be anything "beyond" the CMB?

Yet we also have a third axiom:

3. There are some parts of the universe we can never observe because they are receding away from us at a superluminary speed.

This is simply false, so of course it gets you in trouble if you insist on taking it as axiomatic. The average recession velocity goes luminal at redshift $zapprox 1.4$, while there are observed galaxies and objects at $zsim 8$, e.g., Z8 GND 5296 dwarf galaxy and the GRB 090423 gamma ray burst. Additionally, there are at least some candidates for objects much farther than even that, possibly $zsim 12$.

That means that other than the CMB itself, the most distant and most ancient object we might have observed hails from when the universe as around $370,mathrm{Myr}$ old, which is around a thousand times older than the recombination epoch when the universe first became transparent and hence when the cosmic background radiation was emitted. In terms of redshift, the CMB has $zgtrsim 1100$ or so.

But this surely also implies that we can see beyond the CMB if we see anything which has a red shift indicating an expansion speed very close to $c$.

Something at redshift $zgtrsim 1100$ has a recession velocity of $vgtrsim 3.2c$.

This suggests axiom 2 is incorrect - so what should it be instead?

Since (3) says incorrect things, it's best to throw it out instead.