Accurate predictions of inclusive scattering cross sections in the linear response regime require efficient and controllable methods to calculate the spectral density in a strongly-correlated many-body system. In this work we reformulate the recently proposed Gaussian Integral Transform technique in terms of Fourier moments of the system Hamiltonian which can be computed efficiently on a quantum computer. One of the main advantages of this framework is that it allows for an important reduction of the computational cost by exploiting previous knowledge about the energy moments of the spectral density. For a simple model of medium mass nucleus like ⁴⁰Ca and target energy resolution of 1 MeV we find an expected speed-up of ≈ 125 times for the calculation of the giant dipole response and of ≈ 50 times for the simulation of quasi-elastic electron scattering at typical momentum transfers.