I question your saying these receptacles are "non-polarized receptacles." I suspect your client's receptacles are, in fact, polarized. Look at this image of a polarized duplex receptacle:
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The longer slot is required to be connected to the grounded branch circuit conductor. A true "non polarized non grounding receptacle" has two slots that are exactly the same size as each other. A true "non polarized non grounding receptacle" is very rare, in my personal experience, as the NEC was changed in the 1920s to prevent their use.
If your client's receptacles are genuinely "non polarized" your inspector may comment about them, but he can't require them to be wired in a polarized manner. A non polarized receptacle can't be made into something it is not by reversing the branch circuit conductor connections.
As for testing the wiring in this situation, your method of stringing out an insulated conductor that is connected to a known Equipment Grounding Conductor (EGC) is, in my opinion, your first, best method to make quick, accurate measuements of the existing two wire cloth romex wiring method.