Object storage is one of the most widely used storage solutions in cloud computing environments. Replicating data stored in object storage across one or more cloud regions improves reliability and availability while reducing access latency for users. An object storage replication system is required to ensure consistency among replicas while minimizing replication latency. Existing open-source cross-cloud object storage replication systems, such as Skyplane, support object data replication across major cloud platforms but do not guarantee data consistency and exhibit high replication latency under dynamic workloads. To enable low-cost and low-latency object data replication under dynamic workloads, this study proposes a cross-cloud object storage replication system, termed FastRep, based on serverless computing. Data consistency is ensured through object-grained replication task locks and an optimistic replication mechanism. Cloud function invocation latency is reduced via a time-bounded complete tree, while bandwidth degradation is mitigated through adaptive function cloning. Experimental results demonstrate that FastRep achieves low-cost replication of dynamically sized object data with latency at the minute level.