Aggregation of the amyloid-\beta(A\beta) protein has been implicated in Alzheimer's disease (AD). Since, low molecular weight A\beta aggregates are hypothesized to serve as the primary toxic species in AD pathogenesis, significant research has been conducted to understand the mechanistic details of the aggregation process. We previously demonstrated that heterotypic interactions between A\beta and fatty acids (FAs) can lead to competing pathways of A\beta aggregation, termed as the off-pathway; this off-pathway kinetics can also be modulated by FA concentrations as captured by mass action models. We employed ensemble kinetics simulations which uses a system of Ordinary Differential Equations to model the competing on-and off-pathways of A\beta aggregation that were trained and validated by biophysical experiments. However, these models had several rate constants, treated as free parameters to be estimated, which resulted in over-fitting of the model. Hence, in this paper, we present a global fitting based method to accurately identify the rate constants involved in the complex competing pathway model of A\beta aggregation. We additionally employ detailed parameter identifiability tests for uncertainty quantification using the profile likelihood method. Since, the emergence of off-or on-pathway aggregates are typically controlled by a narrow set of rate constants, it is imperative to rigorously identify the proper rate constants involved in these pathways. These rate constants serve as a basis for future experiments on modulating the aggregation pathways to populate a particular possibly less toxic oligomeric species. The obtained rate constants also motivate new biophysical experiments to better understand the mechanisms of amyloid aggregation in other neurodegenerative diseases.