Cellular vehicle-to-everything (V2X) communication is expected to herald the age of autonomous vehicles in the coming years. With the integration of blockchain in such networks, information of all granularity levels, from complete blocks to individual transactions, would be accessible to vehicles at any time. Specifically, the blockchain technology is expected to improve the security, immutability, and decentralization of cellular V2X communication through smart contract and distributed ledgers. Although blockchain-based cellular V2X networks hold promise, many challenges need to be addressed to enable the future interoperability and accessibility of such large-scale platforms. One such challenge is the offloading of mining tasks in cellular V2X networks. While transportation authorities may try to balance the network mining load, the vehicles may select the nearest mining clusters to offload a task. This may cause congestion and disproportionate use of vehicular network resources. To address this issue, we propose a game-theoretic approach for balancing the load at mining clusters while maintaining fairness among offloading vehicles. Keeping in mind the low-latency requirements of vehicles, we consider a finite channel blocklength transmission which is more practical compared to the use of infinite blocklength codes. The simulation results obtained with our proposed offloading framework show improved performance over the conventional nearest mining cluster selection technique.