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The position accuracy of the bottom dead centre (BDC) of a slider is a crucial performance index to measure the quality of a multi-link mechanical press. At present, the research on improving the position accuracy of the BDC mainly focuses on error compensation, and there is a lack of research on the tolerance design of the transmission mechanism according to the accuracy requirements of the BDC. In this paper, the motion output accuracy of the press is characterized by the position error of the BDC of the slider and the inclined angle error of the slider. Based on the loop increment method, the error transmission model of the transmission mechanism of a multi-link mechanical press was established. The key error factors that affect the output accuracy of the press were determined by analysis of error sensitivity. On this basis, a precision synthesis method considering manufacturing cost and BDC precision reliability was proposed. Finally, a Monte Carlo simulation and experiments on a toggle mechanical press were conducted to verify the model. The results show that the proposed error model and optimization method are feasible and effective.