Let $\varepsilon>0$. Since $\varphi$ is usc, Lemma 2.24 shows that there exists a continuous function $v$ on $X$ such that $v\geq\varphi$ and $\int v\nu\leq\int\varphi\nu+\varepsilon$. The result now follows since [BFJ11, Corollary 8.6] yields an $\theta$-psh model function $\psi$ such that $\varphi\leq\psi\leq v+\varepsilon$. ∎