We introduce a variational approach for the quantum inverse scattering method to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake this in a manner which does not rely on any prior knowledge of integrability through the existence of a set of conserved operators. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen–Cooper–Schrieffer theory, and Bose–Einstein condensation. The Hamiltonians considered describe systems with interacting Cooper pairs and a bosonic degree of freedom. We obtain general exact solvability requirements which include seven subcases that have previously appeared in the literature.