In this study, we develop a fractional-order cooperative enzymatic reaction model using the Liouville–Caputo derivative to extend its kinetics. We present the existence and uniqueness of the solutions through the theory of nonlinear functional analysis. We find the numerical solutions of the specified model using the Euler method and the Laplace Adomian decomposition method (LADM). The precision of the considered technique is evaluated with the aid of Levenberg-Marquardt neural network (NN) platform, complemented with regression analysis and error distribution statistics. The data are divided into several sets including training (70%), validation (15%), and testing (15%). The approximate solutions are analyzed graphically using 2D and 3D phase portraits for various fractional orders and reaction parameters with the help of MATLAB R2024a. Furthermore, a comparative analysis of numerical solutions is presented for both integer and fractional order dynamical systems. This approach provides a new way to chemical reactions and presents the dynamics of these reactions in a new look under fractional derivatives. These findings highlight the potential of fractional calculus (FC) as a powerful modeling framework for complex biochemical kinetics. The framework introduced here provides a theoretical foundation that may support future research in optimizing biochemical processes, including applications in metabolic engineering and controlled drug release mechanisms.