Beat and downbeat tracking, jointly referred to as Meter Tracking, is a fundamental task in Music Information Retrieval (MIR). Deep learning models have far surpassed traditional signal processing and classical machine learning approaches in this domain, particularly for Western (Eurogenetic) genres, where large annotated datasets are widely available. These systems, however, perform less reliably on underrepresented musical traditions. Carnatic music, a rich tradition from the Indian subcontinent, is renowned for its rhythmic intricacy and unique metrical structures (tālas). The most notable prior work on meter tracking in this context employed probabilistic Dynamic Bayesian Networks (DBNs). The performance of state-of-the-art (SOTA) deep learning models on Carnatic music, however, remains largely unexplored. In this study, we evaluate two models for meter tracking in Carnatic music: the Temporal Convolutional Network (TCN), a lightweight architecture that has been successfully adapted for Latin rhythms, and Beat This!, a transformer-based model designed for broad stylistic coverage without the need for post-processing. Replicating the experimental setup of the DBN baseline on the Carnatic Music Rhythm (CMR$_f$) dataset, we systematically assess the performance of these models in a directly comparable setting. We further investigate adaptation strategies, including fine-tuning the models on Carnatic data and the use of musically informed parameters. Results show that while off-the-shelf models do not always outperform the DBN, their performance improves substantially with transfer learning, matching or surpassing the baseline. These findings indicate that SOTA deep learning models can be effectively adapted to underrepresented traditions, paving the way for more inclusive and broadly applicable meter tracking systems.