Purpose: Accurate assessment of surgical complexity is essential in Laparoscopic Cholecystectomy (LC), where severe inflammation is associated with longer operative times and increased risk of postoperative complications. The Parkland Grading Scale (PGS) provides a clinically validated framework for stratifying inflammation severity; however, its automation in surgical videos remains largely unexplored, particularly in realistic scenarios where complete videos must be analyzed without prior manual curation. Methods: In this work, we introduce STC-Net, a novel framework for SingleTimestamp-based Complexity estimation in LC via the PGS, designed to operate under weak temporal supervision. Unlike prior methods limited to static images or manually trimmed clips, STC-Net operates directly on full videos. It jointly performs temporal localization and grading through a localization, window proposal, and grading module. We introduce a novel loss formulation combining hard and soft localization objectives and background-aware grading supervision. Results: Evaluated on a private dataset of 1,859 LC videos, STC-Net achieves an accuracy of 62.11% and an F1-score of 61.42%, outperforming non-localized baselines by over 10% in both metrics and highlighting the effectiveness of weak supervision for surgical complexity assessment. Conclusion: STC-Net demonstrates a scalable and effective approach for automated PGS-based surgical complexity estimation from full LC videos, making it promising for post-operative analysis and surgical training.