I believe I've found the solution to my problem. It's actually very simple.
At time of emission $t_{em}$, the length traveled by light from the Big Bang is given by (4) above. This length defines the causal patch on the CMB sphere. But then space is expending. At time of observation $t_{ob}$, that length becomes dilated: $$ R_{causal}(t_{ob}) = \frac{a(t_{ob})}{a(t_{em})} \, R_{causal}(t_{em}) = 3 \, t_{ob}^{2/3} \, t_{em}^{1/3}. $$ Then, at time of observation, the radius angle sustained by a patch of causality is $$ \alpha \approx \frac{R_{causal}(t_{ob})}{\mathcal{D}(t_{ob})} = \frac{1}{(t_{ob}/t_{em})^{1/3} - 1} \approx 1,64^{\circ}. $$