  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 non-centrosymmetry,
such as an increased spin susceptibility in spin-singlet superconductors for
$T\rightarrow 0$, persist even into the quasi-three-dimensional regime. This
leads to a direction dependent spin response which allows 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.
