= Solution
Project from the north pole $N=(0,0,1)$ onto the equatorial plane, identifying that plane with the <complex plane>. The line from $N$ to $P=(X,Y,Z)$ meets the plane in
$$
\boxed{\phi(P)=\frac{X+iY}{1-Z}},\qquad \phi(N)=\infty.
$$
This is <stereographic projection>. Its inverse is
$$
P(u)=\frac1{1+|u|^2}\bigl(2\operatorname{Re}u,2\operatorname{Im}u,|u|^2-1\bigr).
$$
Rotation through $\theta$ about the $z$-axis changes $X+iY$ to $e^{i\theta}(X+iY)$ and preserves $Z$, so it becomes the <Möbius transformation> $u\mapsto e^{i\theta}u$.
Let $B$ be the rotation <matrix> supplied in the question. It sends the $z$-axis to the $x$-axis, and therefore $R_x(\theta)=BR_z(\theta)B^{-1}$. If $b=\phi B\phi^{-1}$ is its given <Möbius transformation>, then
$$
\phi R_x(\theta)\phi^{-1}=b\circ(u\mapsto e^{i\theta}u)\circ b^{-1},
$$
which is again a <Möbius transformation>. Such $z$- and $x$-axis rotations can send any chosen point of the sphere to the south pole: first rotate its horizontal projection onto the $y$-axis, then rotate in the $yz$-plane. This supplies the rotations needed below.
The <antipodal stereographic coordinate relation> is $\phi(-P)=-1/\overline{\phi(P)}$, as follows immediately from the inverse formula. Fix the <cross-ratio> convention
$$
[a,b;c,d]=\frac{(a-c)(b-d)}{(a-d)(b-c)}.
$$
We claim the required ordering is
$$
\boxed{\left[u,-\frac1{\bar u};v,-\frac1{\bar v}\right]=-\tan^2(d/2)}.
$$
For the <antipodal cross-ratio and spherical distance> identity, rotate $P$ to the south pole. Its coordinate becomes zero and its antipode becomes infinity. A point at angular <spherical distance> $d$ from the south pole has vertical coordinate $-\cos d$ and horizontal magnitude $\sin d$, so its new projected coordinate $v'$ has modulus $\sin d/(1+\cos d)=\tan(d/2)$. By <Möbius invariance of the cross-ratio>, the displayed expression equals
$$
[0,\infty;v',-1/\bar v']=\frac{-v'}{1/\bar v'}=-|v'|^2=-\tan^2(d/2).
$$
For finite nonsingular coordinates the same ratio is $-|u-v|^2/|1+u\bar v|^2$. Zero or infinite antipodal coordinates are interpreted by limits. When $Q=-P$, $d=\pi$ and the ratio has the extended value infinity; the finite real formula applies for $0<d<\pi$.
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