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Inspired by observations of sunspots embedded in active regions, it is often assumed that large-scale, strong magnetic flux emerges from the Sun's deep interior in the form of arched, cylindrical structures, colloquially known as flux tubes. Here, we continue to examine the different dynamics encountered when these structures are considered as concentrations in a volume-filling magnetic field rather than as isolated entities in a field-free background. Via 2.5D numerical simulations, we consider the buoyant rise of magnetic flux concentrations from a radiative zone through an overshooting convection zone that self-consistently (via magnetic pumping) arranges a volume-filling large-scale background field. This work extends earlier papers that considered the evolution of such structures in a purely adiabatic stratification with an assumed form of the background field. This earlier work established the existence of a bias that created an increased likelihood of successful rise for magnetic structures with one (relative) orientation of twist and a decreased likelihood for the other. When applied to the solar context, this bias is commensurate with the solar hemispherical helicity rules (SHHR). This paper establishes the robustness of this selection mechanism in a model incorporating a more realistic background state, consisting of overshooting convection and a turbulently-pumped mean magnetic field. Ultimately, convection only weakly influences the selection mechanism, since it is enacted at the initiation of the rise, at the edge of the overshoot zone. Convection does however add another layer of statistical fluctuations to the bias, which we investigate in order to explain variations in the SHHR.