Would a classical simulation of an entangled quantum system, using a physical computing device built up out of aggregation of parts in space, have the needed type of nonspatial complexity?

For me, the nonspatiality of an entangled state is that it cannot be identified with a logical conjunction of spatially localized substates. For example, if the individual qubits are spatially localized, a two-qubit state like |01> resolves into a |0> over here and a |1> over there - "spatially localized substates" - whereas a state like |01>+|10> cannot be resolved in that way. If you are simulating the two-qubit state by using, let us say, four spatially localized registers to represent the amplitudes of a general two-qubit state ( c00 |00> + c01 |01> + c10 |10> + c11 |11> ), then you have indeed eliminated the nonspatiality.