Cross-chain bridges constitute the single largest vector of systemic risk in Decentralized Finance (DeFi), accounting for over \$2.8 billion in losses since 2021. The fundamental vulnerability lies in the binary nature of existing bridge security models: a bridge is either fully operational or catastrophically compromised, with no intermediate state to contain partial failures. We present ASAS-BridgeAMM, a bridge-coupled automated market maker that introduces Contained Degradation: a formally specified operational state where the system gracefully degrades functionality in response to adversarial signals. By treating cross-chain message latency as a quantifiable execution risk, the protocol dynamically adjusts collateral haircuts, slippage bounds, and withdrawal limits. Across 18 months of historical replay on Ethereum and two auxiliary chains, ASAS-BridgeAMM reduces worst-case bridge-induced insolvency by 73% relative to baseline mint-and-burn architectures, while preserving 104.5% of transaction volume during stress periods. In rigorous adversarial simulations involving delayed finality, oracle manipulation, and liquidity griefing, the protocol maintains solvency with probability $>0.9999$ and bounds per-epoch bad debt to $<0.2%$ of total collateral. We provide a reference implementation in Solidity and formally prove safety (bounded debt), liveness (settlement completion), and manipulation resistance under a Byzantine relayer model.