Cross-chain transfer facilitates seamless value flow across heterogeneous blockchains by securely pegging source and target token pairs at a 1:1 ratio. Cryptocurrency-backed assets (CBAs) schemes ensure asset-level pegging via over-collateralization, but suffer from low capital efficiency. On-chain light client schemes guarantee pegging via ledger-level validation but exhibit poor cross-chain compatibility. In this paper, we adopt a slow-approval, fast-exit transfer model that binds epoch-based issuance to sufficient collateral and enables timely redemption against malicious over-issuance, ensuring fairness even in the absence of atomicity. This relaxed assumption decouples collateral requirements from total issuance volume and improves capital efficiency. To address the limited interoperability of conventional consensus-based approvals, we further propose a stake-driven, consensus-agnostic approval mechanism. This mechanism reduces overhead by introducing unbiased sampling via epoch-bound randomness and compressing the sampled votes of a single validator using a count-attested Merkle tree. Analysis shows that our model ensures incentive compatibility and sampling correctness, preserving the fairness of cross-chain transfers. Experimental results demonstrate that our approach incurs approximately 3 times the cost of standard single-chain transfers, while maintaining lower validation overhead than existing light client solutions, provided the epoch length exceeds 1.2 days.