We introduce new algorithms to identify the trade-off (TRO) between adopting a distributional belief and hedging against ambiguity when modeling uncertainty in a capacitated fixed charge facility location problem (CFLP). We first formulate a TRO model for the CFLP (TRO-CFLP), which determines the number of facilities to open by minimizing the fixed establishment cost and the maximum expected operational cost evaluated over distributions within a TRO set. This set is defined by an empirical distribution, an ambiguity set, and a TRO parameter that controls the trade-off between solving the TRO-CFLP under the empirical distribution and the worst-case distribution. The TRO-CFLP model enables decision-makers to explore a spectrum of location decisions, from optimistic to conservative. We propose a Spectrum Search Algorithm that identifies the full set of distinct optimal solutions across the TRO parameter space. We also develop a hybrid column-and-constraint generation (hC&CG) algorithm for solving the TRO-CFLP model with a fixed TRO parameter. We employ hC&CG as a subroutine within the Spectrum Search Algorithm. Numerical results demonstrate the computational efficiency of the Spectrum Search Algorithm, the superior performance of hC&CG over state-of-the-art methods, and the practical value of adopting solutions on the TRO-CFLP’s spectrum over those obtained using traditional models.