We propose a partitioned numerical framework for the simulation of fluid–structure interaction problems in fast transient regimes, where strong compressibility effects and shock waves propagate across moving interfaces and may penetrate the solid domain. The method is designed to combine arbitrary-order time integration, sub-cycling capabilities, and strict discrete conservation properties within a unified formulation. The coupling strategy is formulated at the algebraic level as a saddle-point problem that enforces the conservation of mass, momentum, and total energy at the fluid–structure interface. Under standard definiteness assumptions inherited from the underlying fluid and solid discretizations, the resulting coupled system is well posed and admits a unique solution. This formulation also naturally accommodates sliding interface conditions. High-order temporal accuracy is achieved by enforcing the coupling conditions consistently at each Runge–Kutta stage. Sub-cycling enables the fluid and solid solvers to evolve according to their respective stability constraints, without resorting to temporal interpolation at the interface. The present work focuses on second-order spatial accuracy, which is sufficient to assess the proposed coupling and time-integration strategy and remains independent of the specific spatial discretization. A series of numerical experiments in one, two, and three space dimensions demonstrates that the method preserves the expected order of accuracy in time, maintains conservation up to machine precision, and remains robust in shock-dominated configurations involving strong fluid–structure interactions. Overall, these results establish a general and systematic framework for the construction of high-order, conservative, partitioned fluid–structure interaction methods suited to fast dynamics and disparate time scales.