We investigate the hard scatterings of hadronic matrix elements corresponding to hadronic gravitational form factors (GFFs) of the pion and proton using QCD factorization, applying conformal field theory (CFT) tools. These GFFs are key to understanding quark and gluon angular momentum via their connection to DVCS moments. The core object is the non-Abelian \( TJJ \) 3-point function, which shows an anomaly-induced dilaton exchange in the \( t \)-channel. We analyze quark, ghost, and gauge-fixing effects through a CFT-based decomposition and propose a parameterization useful for future DVCS studies at the Electron-Ion Collider. The dilaton interaction is interpolated by a conformal anomaly form factor, defined in the nonconformal case, which is constrained by a (dilaton) sum rule.