= Solution
At fixed <conformal time> and fixed radial coordinate, the metric gives transverse physical separation $D=a(\tau_1)r_1\theta$ to first order in the angle. Thus the <angular diameter distance> is
$$
\boxed{\theta\simeq\frac{D}{a(\tau_1)r_1},\qquad d_A=a(\tau_1)r_1.}
$$
The coordinate $r_1$ is the transverse comoving radius appearing in the angular metric coefficient. It is not the radial geodesic distance $r_{\rm curv}\operatorname{arsinh}(r_1/r_{\rm curv})$; replacing one by the other would omit the curvature dependence of angular projection.
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