Large language models can produce fluent judgments for clinical natural language inference, yet they frequently fail when the decision requires the correct inferential schema rather than surface matching. We introduce CARENLI, a compartmentalised agentic framework that routes each premise-statement pair to a reasoning family and then applies a specialised solver with explicit verification and targeted refinement. We evaluate on an expanded CTNLI benchmark of 200 instances spanning four reasoning families: Causal Attribution, Compositional Grounding, Epistemic Verification, and Risk State Abstraction. Across four contemporary backbone models, CARENLI improves mean accuracy from about 23% with direct prompting to about 57%, a gain of roughly 34 points, with the largest benefits on structurally demanding reasoning types. These results support compartmentalisation plus verification as a practical route to more reliable and auditable clinical inference.