Conformal regression provides finite-sample marginal coverage, but it does not by itself determine how interval width should adapt across heterogeneous inputs. Existing locally adaptive methods mainly account for aleatoric noise, leaving uncertainty from weak training support less explicit. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), a split conformal regression method that uses heteroscedastic last-layer Laplace uncertainty as the local normalization scale. CLAPS combines learned input-dependent noise with last-layer epistemic uncertainty, while retaining validity through standard conformal calibration. We characterize this aleatoric--epistemic scale, derive its heteroscedastic last-layer precision, and show that it reduces to aleatoric local scaling as epistemic uncertainty contracts. Experiments show nominal-level coverage with competitive interval efficiency.