The effect of non-demographic noise on stochastic populations has been extensively studied in recent years. For example, various types of non-demographic noise, such as bursty reproduction processes with an a-priori unknown burst size, or time-varying environments, have been separately found to exponentially alter the extinction time of isolated populations. In this work we investigate the extinction of an isolated population under the combined effect of these two types of non-demographic noise. Using a momentum-space WKB approach we arrive at a set of time-dependent Hamilton equations. In order to account for the time-dependency, we find the instanton of a time-perturbed Hamiltonian, which can be analytically found in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically-varying rates corresponding to seasonal effects, and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.