Disaggregated memory (DM) is a promising data center architecture that decouples CPU and memory into independent resource pools to improve resource utilization. Building on DM, memory-disaggregated key-value (KV) stores are adopted to efficiently manage remote data. Unfortunately, existing approaches suffer from poor performance due to two critical issues: 1) the overdependence on one-sided atomic operations in index processing, and 2) the constrained efficiency in compute-side caches. To address these issues, we propose FlexKV, a memory-disaggregated KV store with index proxying. Our key idea is to dynamically offload the index to compute nodes, leveraging their powerful CPUs to accelerate index processing and maintain high-performance compute-side caches. Three challenges have to be addressed to enable efficient index proxying on DM, i.e., the load imbalance across compute nodes, the limited memory of compute nodes, and the expensive cache coherence overhead. FlexKV proposes: 1) a rank-aware hotness detection algorithm to continuously balance index load across compute nodes, 2) a two-level CN memory optimization scheme to efficiently utilize compute node memory, and 3) an RPC-aggregated cache management mechanism to reduce cache coherence overhead. The experimental results show that FlexKV improves throughput by up to 2.94$\times$ and reduces latency by up to 85.2%, compared with the state-of-the-art memory-disaggregated KV stores.