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We study the process, within classical general relativity, in which an incident gravitational plane wave, of weak amplitude and long wavelength, scatters off a massive spinning compact object, such as a black hole or neutron star. The amplitude of the asymptotic scattered wave, considered here at linear order in Newton's constant $G$ while at higher orders in the object's multipole expansion, is a valuable characterization of the response of the object to external gravitational fields. This amplitude coincides with a classical ($\hbar\to0$) limit of a quantum 4-point (object and graviton in, object and graviton out) gravitational Compton amplitude, at the tree (linear-in-$G$) level. Such tree-level Compton amplitudes are key building blocks in generalized-unitary-based approaches to the post-Minkowskian dynamics of binaries of spinning compact objects. In this paper, we compute the classical amplitude using an effective worldline theory to describe the compact object, determined by an action functional for translational and rotational degrees of freedom, including couplings of spin-induced higher multipole moments to spacetime curvature. We work here up to the levels of quadratic-in-spin quadrupole and cubic-in-spin octupole couplings, respectively involving Wilson coefficients $C_2$ and $C_3$. For the special case $C_2=C_3=1$ corresponding to a black hole, we find agreement through cubic-in-spin order between our classical amplitude and previous conjectures arising from considerations of quantum scattering amplitudes. We also present new results for general $C_2$ and $C_3$, anticipating instructive comparisons with results from effective quantum theories.