Task offloading management in 6G vehicular networks is crucial for maintaining network efficiency, particularly as vehicles generate substantial data. Integrating secure communication through authentication introduces additional computational and communication overhead, significantly impacting offloading efficiency and latency. This paper presents a unified framework incorporating lightweight Identity-Based Cryptographic (IBC) authentication into task offloading within cloud-based 6G Vehicular Twin Networks (VTNs). Utilizing Proximal Policy Optimization (PPO) in Deep Reinforcement Learning (DRL), our approach optimizes authenticated offloading decisions to minimize latency and enhance resource allocation. Performance evaluation under varying network sizes, task sizes, and data rates reveals that IBC authentication can reduce offloading efficiency by up to 50% due to the added overhead. Besides, increasing network size and task size can further reduce offloading efficiency by up to 91.7%. As a countermeasure, increasing the transmission data rate can improve the offloading performance by as much as 63%, even in the presence of authentication overhead. The code for the simulations and experiments detailed in this paper is available on GitHub for further reference and reproducibility [1].