We explore the statistical properties of energy transfer in ensembles of doubly-driven Random- Matrix Floquet Hamiltonians, based on universal symmetry arguments. The energy pumping efficiency distribution P(E) is associated with the Hamiltonian parameter ensemble and the eigenvalue statistics of the Floquet operator. For specific Hamiltonian ensembles, P(E) undergoes a transition that cannot be associated with a symmetry breaking of the instantaneous Hamiltonian. The Floquet eigenvalue spacing distribution indicates the considered ensembles constitute generic nonintegrable Hamiltonian families. As a step towards Hamiltonian engineering, we develop a machine-learning classifier to understand the relative parameter importance in resulting high conversion efficiency. We propose Random Floquet Hamiltonians as a general framework to investigate frequency conversion effects in a new class of generic dynamical processes beyond adiabatic pumps.