Data-driven surveys show that drivers do not always choose the shortest-path for their travels. The ideas of bounded rationality have been used to model this behavior, and relax the main assumption of travel time minimization of the User Equilibrium principle. In this paper, we propose an extension of an existing dynamic traffic assignment framework, for aggregated traffic models based on the Macroscopic Fundamental Diagram and regional networks, that extends the principle of the User Equilibrium to account for bounded rational drivers. We focus on drivers with indifferent preferences, and with preferences for more reliable travel times. The network equilibrium is calculated through Monte Carlo simulations and the classical Method of Successive Averages. We first investigate how the drivers' preferences for reliable travel times influences the traffic dynamics in the regional network. We then discuss a potential application example of the proposed methodological framework for estimating the emissions of Carbon Dioxide CO2 and Monoxide NOx at the network level. The results shed light on the importance of properly accounting for more realistic drivers' behavior for estimating emissions. The main contributions of this study lie on the edge between the disciplines of traffic flow theory and network modeling, with a great potential of application for practitioners to assess traffic emissions on large metropolitan areas.