Institutions may benefit from collaborative inference on time-series data. In settings where privacy is necessary, multi-party computation (MPC) is a straightforward approach to providing strong guarantees, yet it remains prohibitively expensive and scales poorly with modern transformer architectures. Vertical Federated Learning (VFL) offers efficiency but suffers from privacy leakage at the embedding level, and securing the entire VFL model head via MPC remains prohibitively slow and communication-heavy for larger models. To enable practical, secure inference at scale, we propose "Public/Private Hybrid Head-VFL" (PPHH-VFL). This hybrid architecture splits the model head into an efficient plaintext public head and a secure, lightweight MPC private head. By applying adversarial training to the public embeddings, we mitigate privacy leakage; concurrently, the small private head securely preserves the flow of sensitive information needed for high downstream utility. Empirical evaluations on models ranging up to 86 million parameters demonstrate that PPHH-VFL accelerates inference by up to six orders of magnitude compared to end-to-end MPC. Compared to a standard VFL+MPC baseline, our approach scales significantly better, achieving a speedup of up to 44.4x in WAN and a 91.2x reduction in communication costs (dropping from 1.7 GB to 19 MB per batch), while simultaneously improving downstream classification accuracy by 2.50% and regression RMSE by 40.7%.