Compute connection 1-forms of warped product manifold using method of moving frames

Your instincts are right. What you're missing is the usual symmetry/skew-symmetry argument combining the Cartan lemma with skew-symmetry of the Levi-Civita connection forms to prove uniqueness. Recall that if $\sum \omega^i\wedge \omega^j_i = 0$, then $\omega^j_i = \sum a^j_{ik}\omega^k$, with $a^j_{ik}=a^j_{ki}$ by the Cartan lemma. But the fact that these are connection forms for the Levi-Civita connection tells you that $a^j_{ik} = -a^i_{jk}$. Thus, $$a^j_{ik} = -a^i_{jk} = -a^i_{kj} = a^k_{ij} = a^k_{ji} = -a^j_{ki} = -a^j_{ik},$$ so $\omega^j_i = 0$, as you wish.