= Solution
Evaluate the metric <inner products> of the proposed <orthonormal tetrad> to first order in the <vector cosmological perturbation>:
$$
\begin{aligned}
g(E_0,E_0)&=-1,\\
g(E_0,E_i)&=a^{-2}(g_{00}B_i+g_{0i})=-B_i+B_i=0,\\
g(E_i,E_j)&=a^{-2}(g_{ij}+B_i g_{0j}+B_j g_{i0}+B_iB_jg_{00})=\delta_{ij}+O(B^2).
\end{aligned}
$$
The <orthonormal tetrad> therefore has <metric signature> $(-,+,+,+)$ to the required order. Put $q=\epsilon/a^2$ and $b=\mathbf e\cdot\mathbf B$. The <photon> momentum components are
$$
p^0=q(1+b),\qquad p^i=qe^i.
$$
They satisfy the <null vector> condition to first order, with $|\mathbf e|=1$.
Use $\eta$ as parameter in the time component of the <geodesic equation>:
$$
\frac{dp^0}{d\eta}+\Gamma^0_{\alpha\beta}\frac{p^\alpha p^\beta}{p^0}=0.
$$
The derivative of the direction is first order, since the background <photon> moves on a straight comoving line; $B_i\,de^i/d\eta$ is consequently second order. With $d\mathbf x/d\eta=\mathbf e+O(B)$,
$$
\frac1q\frac{dp^0}{d\eta}=\left(\frac{\epsilon'}\epsilon-2\mathcal H\right)(1+b)+e^i\dot B_i+e^ie^j\partial_jB_i+O(B^2).
$$
The supplied <Levi-Civita connection> coefficients give
$$
\frac1q\Gamma^0_{\alpha\beta}\frac{p^\alpha p^\beta}{p^0}=2\mathcal H(1+b)-e^ie^j\partial_jB_i+O(B^2).
$$
The spatial gradients and all background expansion terms cancel. Since $\epsilon'/\epsilon$ vanishes in the background, its product with $b$ is also second order. The <photon energy redshift from a vector metric perturbation> is therefore
$$
\boxed{\frac{d\ln\epsilon}{d\eta}+e^i\dot B_i=0.}
$$
An independent check uses the covariant component $p_0=g_{0\mu}p^\mu=-\epsilon+O(B^2)$. The covariant <geodesic equation> gives $dp_0/d\eta=(2p^0)^{-1}\partial_\eta g_{\alpha\beta}p^\alpha p^\beta=\epsilon e^i\dot B_i$; the conformal expansion term vanishes by the <null vector> condition and reproduces the same result.
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