Salt-doped polymers without added solvents have the potential to be nonflammable, robust electrolytes for batteries but suffer from low ion conductivity. Using bulky anions with delocalized charge may reduce ion agglomeration and increase conduction. However, size asymmetry between ions may increase preferential solvation of cations versus the larger anions, lowering the transference number t(+) (fraction of conductivity contributed by the cation, relevant to battery performance). We use coarse-grained molecular dynamics simulations, including a 1/r(4) potential to capture size-dependent solvation effects, to relate polymer and ion chemistry to t(+) and overall conductivity. At low ion size disparity, increasing dielectric constant improves cation conduction due to the mitigation of ion aggregation and correlated cation-anion motion. However, at high ion size disparity, high dielectric medium results in strong preferential solvation of cations, reducing cation mobility and t(+). The trade-off between better cation-anion separation and cation preferential solvation suggests that the strategy of developing high dielectric polymers may enhance performance only at low ion size disparity.