We prove classical simulation hardness, under the generalized $𝖯\ne 𝖭𝖯$ conjecture, for quantum circuit families with applications in near-term molecular electronic ground and excited state estimation. The proof exploits a connection to particle number conserving matchgate circuits with fermionic magic state inputs, which are shown to be universal for quantum computation under post-selection, and are therefore not classically simulable in the worst case, in either the strong (multiplicative) or weak (sampling) sense. We apply these results to non-orthogonal quantum eigensolvers designed for near-term hardware by ruling out sample-based dequantization strategies to additively approximate the off-diagonal matrix elements between orbital-rotated correlated reference states that satisfy an irreducible fermionic magic property, which combines anticoncentration in any orbital basis with an average-case multiplicative approximation hardness conjecture for the associated matrix polynomials. We apply this complexity analysis to two choices of ansatz that incorporate both static and dynamic correlations to model the electronic eigenstates of molecular systems. These are linear combinations of orbital-rotated matrix product states, which are preparable in linear depth, and linear combinations of states prepared by generalized UCCSD or UCJ circuits of polynomial depth, for which computing the expectation values of local fermionic observables up to a constant additive error is $𝖡𝖰𝖯$-complete. We discuss the implications for achieving practical quantum advantage in resolving the electronic structure of catalytic systems composed from multivalent transition metal atoms using near-term quantum hardware.