= Solution
Put $u=t-z$, $\ell=du=e^0-e^3$, $p=A'/A$, $q=B'/B$, with primes denoting $d/du$. The <exterior derivatives> are $de^0=de^3=0$, $de^1=p\ell\wedge e^1$, $de^2=q\ell\wedge e^2$. For a <Levi-Civita connection>, the <connection 1 forms> obey both <Cartan's first structure equation> and <metric compatibility>. With lower indices $\omega_{ab}=\eta_{ac}\omega^c{}_b$, the result is
$$
\boxed{\omega_{01}=\omega_{13}=-p\,e^1,\qquad \omega_{02}=\omega_{23}=-q\,e^2,\qquad \alpha=\beta=-\frac{A'}A,\quad \gamma=\delta=-\frac{B'}B.}
$$
All other lower-index forms vanish or follow from $\omega_{ab}=-\omega_{ba}$. Raising the first time index changes its sign: for example $\omega^0{}_1=\omega^1{}_0=p e^1$, whereas $\omega^1{}_3=-p e^1$ and $\omega^3{}_1=p e^1$.
For direct verification, $\omega^1{}_0\wedge e^0+\omega^1{}_3\wedge e^3=-p\ell\wedge e^1=-de^1$, and the same holds for index two. For indices zero and three each term wedges $e^1$ or $e^2$ with itself, so it is zero. These forms are metric-compatible, and uniqueness follows from the <existence and uniqueness of the Levi-Civita connection>. The printed uniqueness statement implicitly includes metric compatibility: the torsion equation alone permits adding nonzero forms $C^a{}_{bc}e^c$ with $C^a{}_{bc}=C^a{}_{cb}$, since their wedge with $e^b$ vanishes.
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