A fully dynamical and finite-temperature study of the electron momentum relaxation rate and mean free path in a coupled system of electrons and bulk LO phonons in a quantum wire structure is presented. Electron-electron and electron-phonon interactions are treated on an equal footing within the leading-order perturbation theory and random-phase approximation. It is demonstrated that coupled-mode effects drastically change the transport properties of the system at low temperatures. In particular, the 'plasmon-like' and 'LO-phonon-like' excitations yield comparable rates which, as a consequence of the singular nature of the 1D density of states, can be large at the threshold. This is in contrast to room temperature results where only the LO-phonon mode contributes significantly to the rate.