Medical Waste Treatment Techniques (MWTT) have become a significant concern due to the imminent risks they pose to human health and the environment. Proper and secure treatment and disposal of toxic and harmful medical waste are essential and various MWTT options are available to achieve this. The selection of the ideal MWT option is a complex and crucial Multi Criteria Decision Making (MCDM) problem as the decision is influenced by several factors both qualitative and quantitative aspects. This study presents a hybrid MCDM method for analyzing and opting the MWT options within a Hyperbolic fuzzy framework. The Hyperbolic Fuzzy Set (HyFS) is an advanced tool that addresses uncertainty with greater precision, providing more flexibility for the decision makers. An entropy measure and a score function have been introduced in a hyperbolic fuzzy environment. Objective weights are evaluated using the entropy measure while subjective weights are assessed through the Stepwise Weight Assessment Ratio Analysis (SWARA) model. Consequently, a pioneering hybrid MCDM approach is presented combining HyF-EM-SWARA with Complex Proportional Assessment (COPRAS) and Additive Ratio Assessment (ARAS) techniques to identify the optimal MWT option in India. Furthermore, relative evaluations and variability analysis are presented to demonstrate the stability and reliability of the proposed hybrid MCDM methods for ranking the preferences of MWTT.