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In this work we first collect and generalize several existing non-perturbative models for the interaction between a single two-level qubit detector and a relativistic quantum scalar field in arbitrary curved spacetimes, where the time evolution is given by simple-generated unitaries, i.e., those generated by Schmidt rank-1 interaction Hamiltonians. We then extend the relativistic quantum channel associated to these non-perturbative models to include a very large class of Gaussian states of the quantum field, that includes an arbitrary combinations of coherent and squeezing operations (i.e., Gaussian operations) on the field. We show that all physical results involving the non-vacuum Gaussian states can be rephrased in terms of interaction with the vacuum state but with Gaussian operators applied to the field operators via the adjoint channel, effectively giving a "Fourier transformed" interpretation of the Gaussian operations in terms of the causal propagators in spacetime. Furthermore, we show that in these non-perturbative models it is possible to perform exact computation of the Rényi entropy and hence, via the replica trick, the von Neumann entropy for the field state after the interaction with the detector, without making any assumptions about the purity of the joint initial states of the detector and the field. This gives us a three-parameter family of "generalized cat states" of the field whose entropies are finite and exactly computable.