We investigate the sensitivity to the Higgs self-coupling through $W$ boson fusion di-Higgs production at CLIC with a center-of-mass energy of $\sqrt{s}=3$~TeV. We study the interplay between the Higgs self-coupling modifier (${\kappa }_{\lambda }$) and Higgs-gauge boson coupling modifiers (${\kappa }_V$ and ${\kappa }_{2V}$) within the $\kappa$ framework. To enhance the separation between signal and background, we develop a graph neural network (GNN) based classifier that achieves a signal significance of $𝒵\approx 20\sigma$ at $5{ab}^{-1}$, substantially exceeding projected HL-LHC sensitivity. Our results demonstrate that high-energy lepton colliders, combined with graph-based machine learning, provide excellent sensitivity to the Higgs self-coupling and offer a powerful probe of new physics in the electroweak sector, potentially disentangling underlying SMEFT-like and HEFT-like structures.