We present two approaches for obtaining eigensolutions of periodic time-dependent Hamiltonians. Assuming a scale separation between two spatial regions where different potentials dominate, in the first approach we derive a new explicit expansion for an important class of problems with mixed cylindrical and spherical symmetry, by matching wavefunctions of a periodic linear drive in the exterior region, to an arbitrary interior potential expanded in spherical waves. We also formulate an extended time-dependent variational approach using nonorthogonal wavefunctions, which is suitable in particular to account for the neglected potentials in the intermediate region of space in the first approach. We focus on the application of the proposed approaches within a comprehensive scheme for describing quasi-bound eigenstates of a quantum particle trapped in the attractive potential of a semiclassical, periodically-driven scatterer. The wavefunctions of the scattered particle are solved self-consistently together with the average periodic response force it exerts on the classical motion of the scatterer. Entangled fluctuations of the interacting system are linearized and diagonalized in terms of parametric quantum oscillators. A physical example is given by the interaction of cold atoms co-trapped with an ion in a Paul trap, a platform which in recent years promises to make feasible the investigation of ion-neutral interactions in the ultracold regime. The time-dependent framework that we develop can be applied to general quantum scattering problems of particles subject to periodic fields.