We investigate the integrability of spinning compact body dynamics at quadratic order in spin in four-dimensional Einstein spacetimes admitting a non-degenerate Killing–Yano tensor. Working within the Mathisson–Papapetrou–Tulczyjew–Dixon framework under the Tulczyjew–Dixon spin supplementary condition, we model the spin-induced quadrupole with a deformability parameter $\kappa$, where $\kappa =1$ corresponds to black holes. The dynamics is formulated as a Hamiltonian system on a 10-dimensional physical phase space obtained by Dirac–Bergmann reduction. For = 1, we establish Liouville–Arnold integrability at quadratic order in spin by constructing five independent, Poisson-commuting first integrals, including a generalization of the Carter constant and the Rüdiger constant to quadratic-in-spin order in Einstein spacetimes beyond Kerr. For $\kappa \ne 1$, the Rüdiger and Carter constants are no longer conserved; integrability does not persist at this order. All our results are carried out in a covariant manner and numerically verified, and Kerr is recovered as a special case. These results show that integrability can extend beyond Kerr and beyond the linear-in-spin regime, while its breakdown for $\kappa \ne 1$ points to the spin-induced quadrupole as a decisive probe of compact body structure.