The electric dipole transitions $\chi_{bJ}(1P)\to \gamma\Upsilon(1S)$ with $J=0,1,2$ and $h_{b}(1P)\to \gamma\eta_{b}(1S)$ are computed using the weak-coupling version of a low-energy effective field theory named potential non-relativistic QCD (pNRQCD). In order to improve convergence and thus give firm predictions for the studied reactions, the full static potential is incorporated into the leading order Hamiltonian; moreover, we must handle properly renormalon effects and re-summation of large logarithms. The precision we reach is $k_{\gamma}^{3}/(mv)^{2} \times \mathcal{O}(v^{2})$, where $k_{\gamma}$ is the photon energy, $m$ is the mass of the heavy quark and $v$ its velocity. Our analysis separates those relativistic contributions that account for the electromagnetic interaction terms in the pNRQCD Lagrangian which are $v^{2}$ suppressed and those that account for wave function corrections of relative order $v^{2}$. Among the last ones, corrections from $1/m$ and $1/m^2$ potentials are computed, but not those coming from higher Fock states since they demand non-perturbative input and are $\Lambda_{\text{QCD}}^{2}/(mv)^{2}$ or $\Lambda_{\text{QCD}}^{3}/(m^{3}v^{4})$ suppressed, at least, in the strict weak coupling regime.