In this work, we examine solute transport through a rough-walled bifurcating fracture over a range of P'{e}clet numbers and for different fluid discharge ratios between fracture arms, using a combination of lattice-Boltzmann methods for fluid flow and finite difference methods for solute transport. Our choice of P'{e}clet numbers allows us to focus on the transport regime as it transitions from a diffusion-dominated to an advection-dominated one, while the variation of discharge ratios highlights the impact of the bifurcation on the solute distribution and mixing behavior. Using different indicators, including solute concentration moments, dispersivity, the flux-related dilution index, and the scalar dissipation rate, we paint a complete picture of the solute transport and mixing behavior. Due to the rough walls of the bifurcating fracture, the fluid flow field exhibits channeling and includes reverse flow at the fracture intersection when one of the fracture arms is a dead end. The highest accumulated solute mass in the fracture can be observed during the transition of the transport regime when fluid discharge is evenly distributed across both fracture arms. Compared to dominant fluid flow in either fracture arm, evenly-distributed fluid flow across the two arms also induces the highest solute mass transfer. As the P'{e}clet~(Pe) number increases, the weakening of the diffusive flux, with respect to the advective flux, intensifies the impact of stagnation zones on the tailing of the solute plume and on the ""undiluted" behavior of solute. The ""undiluted" behavior is indicated by the development of ""negative" dispersivities at later stages of solute transport and is confirmed by the reduction of the flux-related dilution index at the cross-sections of the fracture intersection. We argue that the ""undiluted" behavior is primarily caused by the development of solute plume tails and the release of solute, in the stagnant zones, back to the flow channels. Both the flux-related dilution index and the scalar dissipation rate indicate that an increase in solute advection leads to enhancements in solute mixing, due to the increase in stretching and deformation of the solute plume in such heterogeneous flow fields, particularly under evenly distributed fluid discharge across both fracture arms. The scaling of the dissipation rate against time suggests a minimum scaling slope that is constrained by a Fickian process and a maximum scaling slope that increases with~Pe.