Any faults can/will cause issues. Without knowing the exact how that the system was designed to react to the faults, you can only guess as to what is going on. The downstream O2 sensors are often used in setting fuel trims, so you may be running into issues there, but that is just a guess. Personally, I wouldn't the car that way.
Have you tried flashing the DME with ROW code?
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'03 S, manual, 18" Carrera wheels, PSM, PSE, Litronic, 996 Cluster, +
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