Calculations are presented of n-type carrier mobility in a gated, double-quantum-well InGaAs/GaNxAs1-x heterostructure (with x approximate to 0.01), where transport occurs in the lowest two subbands (one primarily in the InGaAs layer and the other primarily in the dilute nitride layer) and carrier confinement is provided by surrounding AlGaAs layers and gating fields from front and back gates. Such a device allows a controlled interaction between carriers and localized nitrogen cluster states near the conduction band edge in the dilute nitride. Varying the In composition in the GaInAs layer and the gating fields is predicted to alter the Fermi level, relative to the localized nitrogen cluster states, and cause a strong modulation of the n-type mobility. For appropriately designed heterostructures, the mobility is predicted to vary by up to a factor of three when the Fermi level is resonant with nitrogen cluster electronic states. Thus, the mobility measurements in such a heterostructure can be used as a spectroscopic probe of the localized states in the dilute nitride layer.