We may assume that $\|\cdot\|$ is a piecewise linear metric. Let $\mathscr{X}^{\prime}$ be an algebraic model of $X$ on which $(\mathcal{O}_{X^{\textup{an}}},\|\cdot\|)$ has an algebraic model $\mathscr{M}$. The section $1$ of $\mathcal{O}_{X}$ extends to a meromorphic section $s$ of $\mathscr{M}$ and then $\mathscr{M}=\mathcal{O}(D)$ for the vertical Cartier divisor $D=\Div(s)$ on $\mathscr{X}^{\prime}$. By [GW10, Theorem 13.98] we may assume that $\mathscr{X}^{\prime}$ is a vertical blowup of $\mathscr{X}$. Denote by $\pi$ the canonical map $\mathscr{X}^{\prime}\rightarrow\mathscr{X}$. We show first that $D$ is $\pi$-nef, i.e. $\deg(D\cdot C)\geq 0$ for any closed curve $C\subseteq\tilde{\mathscr{X}}^{\prime}$ which is contracted by $\pi$.
So let $x\in\tilde{\mathscr{X}}$ be a closed point and $C\subseteq\pi^{-1}(x)$ a curve. Then by the semipositivity assumption $\deg((\mathcal{O}(D)+\pi^{\ast}\mathscr{L})\cdot C)\geq 0$. But since $\pi_{\ast}(\pi^{\ast}\mathfrak{L}\cdot C)=\mathscr{L}\cdot\pi_{\ast}(C)=0$ we have $\deg(\pi^{\ast}\mathscr{L}\cdot C)=0$ and hence $\deg(D\cdot C)\geq 0$.
Now let $A$ be a $\pi$-ample vertical Cartier divisor on $\mathscr{X}^{\prime}$, e.g. $A=-E$ for the exceptional divisor $E$ of the blowup (this is $\pi$-ample by [GW10, Proposition 13.96]). Then $D+A$ is $\pi$-ample by the relative version of Kleiman’s criterion ([Deb01, Remark 7.41]). Furthermore, since $\mathcal{O}_{\mathscr{X}^{\prime}}(D)$ and $\mathcal{O}_{\mathscr{X}^{\prime}}(A)$ are coherent vertical fractional ideal sheaves, also $\mathfrak{a}:=\pi_{\ast}\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))$ is a coherent vertical fractional ideal sheaf on $\mathscr{X}$ for any $m\in\mathbb{N}_{>0}$ by [Ull95, Theorem 5.3].
By the characterization of $\pi$-ampleness in [Gro61, Proposition 4.6.8] there exists some $m\in\mathbb{N}_{>0}$ such that $\pi^{\ast}\mathfrak{a}\rightarrow\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))$ is surjective. This implies

$$\log|\mathfrak{a}|=\log|\pi^{\ast}\mathfrak{a}|=\log|\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))|=m\cdot(\varphi-\log\|1\|_{\mathcal{O}_{\mathscr{X}^{\prime}}(A)})$$

and hence $\frac{1}{m}\log|\mathfrak{a}|=\varphi-\log\|1\|_{\mathcal{O}_{\mathscr{X}^{\prime}}(A)}$. Since we can replace $A$ by $\epsilon A$ for arbitrary small $\epsilon\in\mathbb{Q}_{>0}$ this concludes the proof. ∎