We establish a factorization relation between baryon quasi-distribution amplitudes (quasi-DAs) defined with gradient flow and their counterparts renormalized in the MS scheme. Working beyond the small flow-time limit, we perform a complete one-loop calculation that yields the full matching kernel and the associated Wilson-line linear divergence for flowed quasi-DAs. Building on this relation, we formulate a hybrid renormalization scheme that combines ratio scheme with gradient-flow matching. The framework enables a systematic connection between lattice QCD matrix elements and continuum baryon quasi-DAs, with reduced renormalization uncertainties and clear guidance for practical lattice analyses.