  We study four-point functions of arbitrary half-BPS operators in a
4-dimensional $\mathcal{N}=2$ SCFT with flavour group $SO(8)$ at genus-zero and
strong 't Hooft coupling, corresponding - via AdS/CFT - to the ($\alpha'$
expansion of the) Veneziano amplitude on an AdS$_5 \times$S$^3$ background. We
adapt a procedure first proposed by Abl, Heslop and Lipstein in the context of
AdS$_5 \times$S$^5$, and postulate the existence of an effective action in
terms of an $8$-dimensional scalar field valued in the adjoint of the flavour
group. The various Kaluza-Klein correlators can then be computed by uplifting
the standard AdS/CFT prescription to the full product geometry with AdS
bulk-to-boundary propagators and Witten diagrams replaced by suitable AdS$_5
\times$S$^3$ versions. After elucidating the main features of the procedure,
valid at all orders in $\alpha'$, we show explicit results up to order
$\alpha'^{5}$. The results provide further evidence of a novel relation between
AdS$\times$S and flat amplitudes - which made its first appearance in
$\mathcal{N}=4$ SYM - that is perhaps the most natural extension of the well
known flat-space limit proposed by Penedones to cases where AdS and S have the
same radius.
