Record matching models typically output a real-valued matching score that is later consumed through thresholding, ranking, or human review. While fairness in record matching has mostly been assessed using binary decisions at a fixed threshold, such evaluations can miss systematic disparities in the entire score distribution and can yield conclusions that change with the chosen threshold. We introduce a threshold-independent notion of score bias that extends standard group-fairness criteria-demographic parity (DP), equal opportunity (EO), and equalized odds (EOD)-from binary outputs to score functions by integrating group-wise metric gaps over all thresholds. Using this metric, we empirically show that several state-of-the-art deep matchers can exhibit substantial score bias even when appearing fair at commonly used thresholds. To mitigate these disparities without retraining the underlying matcher, we propose two model-agnostic post-processing methods that only require score evaluations on an (unlabeled) calibration set. Calib targets DP by aligning minority/majority score distributions to a common Wasserstein barycenter via a quantile-based optimal-transport map, with finite-sample guarantees on both residual DP bias and score distortion. C-Calib extends this idea to label-dependent notions (EO/EOD) by performing barycenter alignment conditionally on an estimated label, and we characterize how its guarantees depend on both sample size and label-estimation error. Experiments on standard record-matching benchmarks and multiple neural matchers confirm that Calib and C-Calib substantially reduce score bias with minimal loss in accuracy.