We consider the Hilbert-type operator defined by $$ H_{\omega}(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We prove that $H_{\omega}$ is bounded from $H^\infty$ to the Bloch space if and only if $\omega$ belongs to the class $\widehat{\mathcal{D}}$, which consists of radial weights $\omega$ satisfying the doubling condition $\sup_{0\le r< 1} \frac{\int_r^1 \omega(s)\,ds}{\int_{\frac{1+r}{2}}^1\omega(s)\,ds}< \infty$. Further, we describe the weights $\omega\in \widehat{\mathcal{D}}$ such that $H_\omega$ is bounded on the Hardy space $H^1$, and we show that for any $\omega\in \widehat{\mathcal{D}}$ and $p\in (1,\infty)$, $H_\omega:\,L^p_{[0,1)} \to H^p$ is bounded if and only if the Muckenhoupt type condition \begin{equation*} \sup\limits_{0< r< 1}\left(1+\int_0^r \frac{1}{\widehat{\omega}(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 \omega(t)^{p'}\,dt\right)^{\frac{1}{p'}} < \infty, \end{equation*} holds. Moreover, we address the analogous question about the action of $H_{\omega}$ on weighted Bergman spaces $A^p_\nu$.