The rapid growth of Internet of Medical Things (IoMT) devices has resulted in significant security risks, particularly the risk of malware attacks on resource-constrained devices. Conventional deep learning methods are impractical due to resource limitations, while Federated Learning (FL) suffers from high communication overhead and vulnerability to non-IID (heterogeneous) data. In this paper, we propose a split learning (SL) based framework for IoT malware detection through image-based classification. By dividing the neural network training between the clients and an edge server, the framework reduces computational burden on resource-constrained clients while ensuring data privacy. We formulate a joint optimization problem that balances computation cost and communication efficiency by using a game-theoretic approach for attaining better training performance. Experimental evaluations show that the proposed framework outperforms popular FL methods in terms of accuracy (+6.35%), F1-score (+5.03%), high convergence speed (+14.96%), and less resource consumption (33.83%). These results establish the potential of SL as a scalable and secure paradigm for next-generation IoT security.