We present a momentum-dependent reweighting strategy to extend current LHC di-Higgs analyses within the 𝜅 framework and Standard Model effective field theory into the bosonic sector of the Higgs effective field theory (HEFT). Unlike Standard Model effective field theory, where symmetry constraints tightly correlated multi-Higgs processes, HEFT allows for a broader range of momentum-dependent deviations that can substantially impact di-Higgs kinematics and offer a powerful probe of nonlinear Higgs dynamics. We generalize the interpretation of existing experimental analyses by integrating HEFT operators up to chiral dimension 4 into differential Monte Carlo reweighting. We quantify the sensitivity to representative HEFT operators using multidimensional likelihoods for Run 3 and project the reach at the high-luminosity LHC. Particular emphasis is placed on how different exclusive final states, such as and , respond to momentum enhancements and how their complementary event selections drive exclusion limits. We further explore how rare final states, especially four-top production, can provide orthogonal constraints on HEFT-induced modifications, thereby enhancing global sensitivity to new physics effects in the Higgs sector.