The effect of nonlocality on the optical response of metals lies at the
forefront of research in nanoscale physics and, in particular, quantum
plasmonics. In alkali metals, nonlocality manifests predominantly as electron
density spill-out at the metal boundary, and as surface-enabled Landau damping.
For an accurate description of plasmonic modes, these effects need be taken
into account in the theoretical modelling of the material. The resulting modal
frequency shifts and broadening become particularly relevant when dealing with
the strong interaction between plasmons and excitons, where hybrid modes emerge
and the way they are affected can reflect modifications of the coupling
strength. Both nonlocal phenomena can be incorporated in the classical local
theory by applying a surface-response formalism embodied by the Feibelman
parameters. Here, we implement surface-response corrections in Mie theory to
study the optical response of spherical plasmonic--excitonic composites in
core--shell configurations. We investigate sodium, a jellium metal dominated by
spill-out, for which it has been anticipated that nonlocal corrections should
lead to an observable change in the coupling strength, appearing as a
modification of the width of the mode splitting. We show that, contrary to
expectations, the influence of nonlocality on the anticrossing is minimal, thus
validating the accuracy of the local response approximation in strong-coupling
photonics