Disaggregated serving alleviates memory bottlenecks in Large Language Model (LLM) inference but creates a severe communication bottleneck: transmitting high-dimensional Key-Value (KV) caches often dominates time-to-first-token (TTFT). Moreover, reusing caches across heterogeneous models (e.g., base and fine-tuned variants) causes semantic misalignment that accumulates over layers, degrading generation quality. We propose Semantic Cache Distillation (SCD), a loss-constrained framework that replaces raw KV transmission with compact semantic codes. SCD addresses these challenges via two mechanisms: (1) Reuse, which reconstructs most layers from low-rank subspaces to minimize transfer cost, and (2) Patch, which predicts normalized inputs at sparse transition layers to truncate error propagation. Empirically, SCD delivers up to 2.65 $\times$ TTFT speedup over the oracle consumer prefill and dominates quantization and selective recomputation baselines on the quality--latency Pareto frontier in bandwidth-constrained regimes, while keeping generation quality within 5\% F1 of the oracle.