The growing popularity of mobility-on-demand fleets increases the importance to understand the impact of mobility-on-demand fleets on transportation networks and how to regulate them. For this purpose, transportation network simulations are required to contain corresponding routing methods. We study the trade-off between computational efficiency and routing accuracy of different approaches to routing fleets in a dynamic network simulation with endogenous edge travel times: a computationally cheap but less accurate Network Fundamental Diagram (NFD) based method and a more typical Dynamic Traffic Assignment (DTA) based method. The NFD-based approach models network dynamics with a network travel time factor that is determined by the current average network speed and scales free-flow travel times. We analyze the different computational costs of the approaches in a case study for 10,000 origin-destination (OD) pairs in a network of the city of Munich, Germany that reveals speedup factors in the range of 100. The trade-off for this is less accurate travel time estimations for individual OD pairs. Results indicate that the NFD-based approach overestimates the DTA-based travel times, especially when the network is congested. Adjusting the network travel time factor based on pre-processed DTA results, the NFD-based routing approach represents a computationally very efficient methodology that also captures traffic dynamics in an aggregated way.