Let me give another proof in the spirit of algebraic geometry.
By Riemann's theorem, for any symmetric theta divisor $\Theta$ there exists a theta characteristic $\kappa$ (i.e., a line bundle such that $\kappa^{\otimes 2}=\omega_C$) on $C$ such that $$W_{g-1}=\Theta + \kappa,$$ where $W_{g-1}$ is the image of the abelian sum mapping $$u \colon \textrm{Sym}^{g-1}(C) \to J(C).$$
$\kappa$ is called the Riemann's constant, and one has
$$\delta^* \Theta = q_1^* (\kappa) \otimes q_2^*(\kappa) \otimes \mathcal{O}_{C^2}(\Delta), \quad \quad (\star)$$
where $q_i \colon C^2 \to C$ are the natural projections.
The proof of such a formula is easy, and it is based on the Seesaw Principle: in other words, one shows that the restrictions to $C \times \{ p \}$ and $\{p \} \times C$ of both sides of $(\star)$ coincide for all $p \in C$.
For the details, see [Birkenhake-Lange, Complex Abelian Varieties, Proposition 11.10.2 $(a)$], putting $\eta=\kappa$ into the statement.
Since $\kappa$ is a theta characteristic it follows that the cohomology class of $q_1^* (\kappa) \otimes q_2^*(\kappa)$ is exactly $\frac{1}{2}(\psi_1 + \psi_2)$. On the other hand, as you noticed, the cohomology class of $\Theta$ is $\phi$, so we are done.