The polarized cross-section ratio σ_{LT^{'}}/σ_{0} from hard exclusive π^{-}Δ^{++} electroproduction off an unpolarized hydrogen target has been extracted based on beam-spin asymmetry measurements using a 10.2 GeV/10.6 GeV incident electron beam and the CLAS12 spectrometer at Jefferson Lab. The study, which provides the first observation of this channel in the deep-inelastic regime, focuses on very forward-pion kinematics in the valence regime, and photon virtualities ranging from 1.5 GeV^{2} up to 7 GeV^{2}. The reaction provides a novel access to the d-quark content of the nucleon and to p→Δ^{++} transition generalized parton distributions. A comparison to existing results for hard exclusive π^{+}n and π^{0}p electroproduction is provided, which shows a clear impact of the excitation mechanism, encoded in transition generalized parton distributions, on the asymmetry.