In this letter, we propose a two-stage design method to construct memory efficient mutual information-maximizing quantized min-sum (MIM-QMS) decoder for rate-compatible low-density parity-check (LDPC) codes. We first develop a modified density evolution to design a unique set of lookup tables (LUTs) that can be used for rate-compatible LDPC codes. The constructed LUTs are optimized based on their discrepancy values and a merge function to reduce the memory requirement. Numerical results show that the proposed rate-compatible MIM-QMS decoder can reduce the memory requirement for decoding by up to 94.92% compared to the benchmark rate-compatible LUT-based decoder with generally faster convergence speed. In addition, the proposed decoder can approach the performance of the floating-pointing belief propagation decoder within 0.15 dB.