A wireless Internet-access mesh network (WIMNET) provides a scalable Internet-access network by flexibly allocating access points (APs) in the field that are connected by wireless links. Previously, we proposed an active AP selection algorithm to select the APs to be activated among the redundantly allocated APs for WIMNET that has a single Internet gateway (GW). However, in practical implementations of large-scale WIMNET, multiple GWs are usually necessary to increase the capacity of Internet connections. Besides, the number of hops between a GW and a host is limited for stable multihop communications, and the transmission speed of a wireless link is dynamically changed by the link condition such as the distance in the high-speed IEEE802.11ac protocol. Furthermore, the minimum throughput should be provided to a host as QoS (Quality of Service) when it is exclusively connected with the GW. In this paper, we propose extensions of the active AP selection algorithm to consider these conditions. We verify the effectiveness of our proposal through simulations in three topologies using the WIMNET simulator.