Using three-dimensional kinetic simulations, we examine the emission of collimated gamma-ray beams from structured laser-irradiated targets with a pre-filled cylindrical channel. The channel guides the incident laser pulse, enabling generation of a slowly evolving azimuthal plasma magnetic field that serves two key functions: to enhance laser-driven electron acceleration and to induce emission of gamma-rays by the energetic electrons. Our main finding is that the conversion efficiency of the laser energy into a beam of gamma-rays ($5^{\circ}$ opening angle) can be significantly increased without increasing the laser intensity by utilizing channels with an optimal density. The conversion efficiency into multi-MeV photons increases roughly linearly with the incident laser power $P$, as we increase $P$ from 1 PW to 4 PW while keeping the laser peak intensity fixed at $5 \times 10^{22}$ W/cm$^2$. This scaling is achieved by using an optimal range of plasma densities in the channel between 10 and $20 n_{cr}$, where $n_{cr}$ is the classical cutoff density for electromagnetic waves. The corresponding number of photons scales as $P^2$. One application that directly benefits from such a strong scaling is the pair production via two-photon collisions, with the number of generated pairs increasing as $P^4$ at fixed laser intensity.