Existing research shows that Cluster-based Medium Access Control (CB-MAC) protocols perform well in controlling and managing Vehicular Ad hoc Network (VANET), but requires ensuring improved security and privacy preserving authentication mechanism. To this end, we propose a multi-level blockchain-based privacy-preserving authentication protocol. The paper thoroughly explains the formation of the authentication centers, vehicles registration, and key generation processes. In the proposed architecture, a global authentication center (GAC) is responsible for storing all vehicle information, while Local Authentication Center (LAC) maintains a blockchain to enable quick handover between internal clusters of vehicle. We also propose a modified control packet format of IEEE 802.11 standards to remove the shortcomings of the traditional MAC protocols. Moreover, cluster formation, membership and cluster-head selection, and merging and leaving processes are implemented while considering the safety and non-safety message transmission to increase the performance. All blockchain communication is performed using high speed 5G internet while encrypted information is transmitted while using the RSA-1024 digital signature algorithm for improved security, integrity, and confidentiality. Our proof-of-concept implements the authentication schema while considering multiple virtual machines. With detailed experiments, we show that the proposed method is more efficient in terms of time and storage when compared to the existing methods. Besides, numerical analysis shows that the proposed transmission protocols outperform traditional MAC and benchmark methods in terms of throughput, delay, and packet dropping rate.