In this talk, we will present a unified Fermi-gas framework for computing BPS observables in ABJM theory on S^3. Supersymmetric localization reduces the problem to an interacting matrix model that can be interpreted as a one-dimensional ideal Fermi gas, whose dynamics is encoded in a set of TBA-like equations. We will show that these equations enable a non-perturbative bootstrap approach that systematically captures the instanton sectors of the free energy and winding Wilson loops. As applications, we will provide an analytic proofs of several previously conjectural formulas and derive closed-form expressions, valid at arbitrary winding number, for the leading membrane- and worldsheet-instanton corrections to the 1/6- and 1/2-BPS  Wilson loops.