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Inversion symmetry is a key symmetry in unconventional superconductors, and even its local breaking can have profound implications. For inversion-symmetric systems, there is a competition on a microscopic level between the spin-orbit coupling associated with the local lack of inversion and hybridizing terms that “restore” inversion. Investigating a layered system with alternating mirror-symmetry breaking, we study this competition considering the spin response of different superconducting order parameters for the case of strong spin-orbit coupling. We find that signatures of the local noncentrosymmetry, such as an increased spin susceptibility in spin-singlet superconductors for T→0, persist even into the quasi-three-dimensional regime. This leads to a direction-dependent spin response that allows us to distinguish different superconducting order parameters. Furthermore, we identify several regimes with possible topological superconducting phases within a symmetry-indicator analysis. Our results may have direct relevance for the recently reported Ce-based superconductor CeRh_{2}As_{2} and beyond.