Why does $mathrm{d}phi' = mathrm{d}phi$ in the derivation of Lorentz transformation of Solid angles?

Let $O$ and $O’$ be two inertial frames where $O’$ moves with velocity $v$ relative to $O$ along the $+x$ direction. A light beam of intersection angle $theta$ with the $x$ axis in frame $O$ has a different intersection angle $theta’$ with the $x’$ axis in the frame $O’$. The transformation rule reads
$$ costheta’ = frac{costheta – beta}{1-betacostheta},quad text{where} quad beta = frac{v}{c}. $$
(For derivations, see Relativistic beaming/ Aberration effect derivation, for example, which is a simple application of the Lorentz transformation.)

Now if we have a collection of light beams that radiate from $mathbf{r}=boldsymbol{0}$ to a solid angle $mathrm{d}Omega$ in the frame $O$, we know that these light beams radiate to a different solid angle $mathrm{d}Omega’$ in the frame $O’$ (because $costheta$ changes to $costheta’$).

We may find $mathrm{d}Omega’$ in terms of $mathrm{d}Omega$ as follows (for simplicity we assume that $mathrm{d}Omega$ lies at $phi = 0$, i.e. the light beams are roughly near the $xz$ plane):
$$begin{align*}
mathrm{d}Omega’ &= mathrm{d}costheta’ mathrm{d}phi’ = frac{mathrm{d} costheta’}{d costheta}frac{mathrm{d}phi’}{mathrm{d}phi}, mathrm{d}costheta ,mathrm{d}phi
&= frac{1-beta^2}{(1-betacostheta)^2}cdot frac{mathrm{d}phi’}{mathrm{d}phi} cdot mathrm{d}Omega
&= frac{1-beta^2}{(1-betacostheta)^2} mathrm{d}Omega.
end{align*}$$
(See, for example, page 47 of this pdf, page 28 of this pdf, or page 13 or this pdf, which all arrive at exactly the same expression except for the different signs before $betacostheta$ in the denominator caused by a different choice of the moving direction of $O’$ relative to $O$.)

The above derivation looks all good except for the last step where we assumed that $displaystyle frac{mathrm{d}phi’}{mathrm{d}phi} = 1$. Some authors argued that ${mathrm{d}phi’} ={mathrm{d}phi}$ because the line segment of $mathrm{d}{phi}$ is perpendicular to $x$, the direction of moving, thus invariant under Lorentz transformations.

However, this statement doesn’t sound quite right to me. What is invariant under the Lorentz transformations should be $mathrm{d}p_y$ and $mathrm{d}p_z$, i.e. the $y$ and $z$ components of the momenta of the light beams. But $mathrm{d}phi$, defined by (remember that we have assumed $phi rightarrow 0$)
$$ mathrm{d}phi = frac{mathrm{d}p_y}{p_x}, $$
does change under Lorentz transformation as
$$ frac{mathrm{d}phi’}{mathrm{d}phi} = frac{p_x}{p_x’} neq 1, $$
which invalidates the above derivation.

The final expression of $mathrm{d}Omega’$ appears in much literature and must be correct. Therefore, I must have missed some points in the proof. Could someone explain why ${mathrm{d}phi’} ={mathrm{d}phi}$?