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 $$W_{g-1}=\Theta + \kappa.$$
$\kappa$ is called the *Riemann's constant*. Therefore the following formula holds:


$$\delta^* \Theta = q_1^* (\kappa) \otimes q_2^*(\kappa) \otimes \mathcal{O}_{C^2}(\Delta), \quad \quad (\star)$$ 

where $q_i  \colon C^2 \to C$ are the natural projections.

The proof is simple, and it is based on the Seesaw Principle: in other words, it is sufficient to show 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.