The rate constant matrix contraction (RCMC) method, proposed by Sumiya et al. (2015, 2017), enables efficient and stable simulations of chemical kinetics in large-scale reaction networks. Iwata et al. (2023) reformulated the RCMC method as an algorithm for solving master equations with rate constant matrices satisfying detailed balance. This study focuses on accelerating RCMC by addressing its bottleneck: the greedy selection of steady states, which is equivalent to MAP inference in determinantal point processes (DPPs) under cardinality constraints. LazyFastGreedy, introduced by Hemmi et al. (2022), combines the practically efficient lazy greedy method with the DPP MAP inference but suffers from numerical instability due to wide-ranging reaction time scales in chemical kinetics. We propose modified LazyFastGreedy that mitigates numerical instability by avoiding subtraction of like-sign numbers, leveraging properties of rate constant matrices and connections to Cholesky decomposition. Numerical experiments on real chemical reaction instances demonstrate that the proposed method achieves significant speed improvements while maintaining numerical stability.