In the realm of enterprise blockchain systems, robust access control is crucial for enabling secure exchange of data and assets across networks. In this paper, we introduce InterAcct, designed to address the intricacies of access control for cross-chain interactions between permissioned networks. InterAcct enforces access control at the source within the originating network as well as for ledger data and assets in the destination network after validating cross-consortium requests. We implement InterAcct using Hyperledger Cacti as the data interoperability layer across Hyperledger Fabric ledgers. We establish that InterAcct handles cross-chain access control for request-responses with less than 2 second latency overhead while adapting to increasing workloads.