Log-Structured Merge-tree-based Key-Value Stores (LSM-KVS) have been optimized and redesigned for disaggregated storage via techniques such as compaction offloading to reduce the network I/Os between compute and storage. However, the constrained memory space and slow flush at the compute node severely limit the overall write throughput of existing optimizations. In this paper, we propose O3-LSM, a fundamental new LSM-KVS architecture, that leverages the shared Disaggregated Memory (DM) to support a three-layer offloading, i.e., memtable Offloading, flush Offloading, and the existing compaction Offloading. Compared to the existing disaggregated LSM-KVS with compaction offloading only, O3-LSM maximizes the write performance by addressing the above issues. O3-LSM first leverages a novel DM-Optimized Memtable to achieve dynamic memtable offloading, which extends the write buffer while enabling fast, asynchronous, and parallel memtable transmission. Second, we propose Collaborative Flush Offloading that decouples the flush control plane from execution and supports memtable flush offloading at any node with dedicated scheduling and global optimizations. Third, O3-LSM is further improved with the Shard-Level Optimization, which partitions the memtable into shards based on disjoint key-ranges that can be transferred and flushed independently, unlocking parallelism across shards. Besides, to mitigate slow lookups in the disaggregated setting, O3-LSM also employs an adaptive Cache-Enhanced Read Delegation mechanism to combine a compact local cache with DM-assisted memtable delegated read. Our evaluation shows that O3-LSM achieves up to 4.5X write, 5.2X range query, and 1.8X point lookup throughput improvement, and up to 76% P99 latency reduction compared with Disaggregated-RocksDB, CaaS-LSM, and Nova-LSM.