Multi-agent LLM decision systems for portfolio management still lack a principled way to assign credit across specialist agents, remain vulnerable to cold-start dominance under regime shifts, and offer limited transparency into how final allocations are formed. We propose Market Regime Council (MRC), a cooperative multi-agent decision system that computes exact Shapley credits across all single, pairwise, and Grand-coalition outputs for online agent weighting. Instantiated with N=3 specialist agents, at each trading period, MRC recomputes coalition-based Shapley weights from exponentially weighted performance histories, uses a Bayesian adaptive mixture to stabilize early periods, applies regime-dependent multipliers to adjust agent authority, and records each rebalance through a five-layer causal trace. Over 1,037 trading days across 13 crypto assets and five seeds, MRC achieves a Sharpe ratio of 1.51 and a cumulative return of 440.1%, ranking first on CR, SR, and IR among active baselines and attaining the lowest MDD among active methods. Ablation results show that the gains come from Shapley-weighted integration across coalition outputs rather than from any single stage in isolation. Code and demo data are included in the supplementary material.