Abstract Active galactic nucleus (AGN) bubbles in cool-core galaxy clusters are believed to facilitate the transport of cosmic-ray electrons (CRe) throughout the cluster. Recent radio observations reveal the complex morphologies of cluster diffuse emission, potentially linked to interactions between AGN bursts and the cluster environment. We perform 3D magnetohydrodynamical simulations of binary cluster mergers and inject a bidirectional jet at the center of the main cluster. Kinetic, thermal, magnetic, and cosmic ray (CR) energy are included in the jet and we use the two-fluid formalism to model the CR component. We explore a wide range of cluster merger and jet parameters. We discuss the formation of various wide-angle-tail and X-shaped sources in the early evolution of the jet and merger. During the last phase of the evolution, we find that the CR material efficiently permeates the central region of the cluster reaching radii of ∼1–2 Mpc within ∼5–6 Gyr, depending on the merger mass ratio. We find that solenoidal turbulence dominates during the binary merger and we explore the possibility for the CR jet material to be reaccelerated by super-Alfvènic turbulence and contribute to cluster scale radio emission. We find high volume fractions, ≳70%, at which the turbulent acceleration time is shorter than the electron cooling time. Finally, we study the merger shock interaction with the CRe material and show that it is unlikely that this material significantly contributes to the radio relic emission associated with the shocks. We suggest that multiple jet outbursts and/or off-center radio galaxies would increase the likelihood of detecting these merger shocks in the radio due to shock reacceleration.