In differential privacy (DP) mechanisms, it can be beneficial to release "redundant" outputs, where some quantities can be estimated in multiple ways by combining different privatized values. Indeed, the DP 2020 Decennial Census products published by the U.S. Census Bureau consist of such redundant noisy counts. When redundancy is present, the DP output can be improved by enforcing self-consistency (i.e., estimators obtained using different noisy counts result in the same value), and we show that the minimum variance processing is a linear projection. However, standard projection algorithms require excessive computation and memory, making them impractical for large-scale applications such as the Decennial Census. We propose the Scalable Efficient Algorithm for Best Linear Unbiased Estimate (SEA BLUE), based on a two-step process of aggregation and differencing that 1) enforces self-consistency through a linear and unbiased procedure, 2) is computationally and memory efficient, 3) achieves the minimum variance solution under certain structural assumptions, and 4) is empirically shown to be robust to violations of these structural assumptions. We propose three methods of calculating confidence intervals from our estimates, under various assumptions. Finally, we apply SEA BLUE to two 2010 Census demonstration products, illustrating its scalability and validity.