Without proper port authentication (802.1x) and/or profiling you will not solve the problem. Using ClearPass for network access control en policy enforcement can help you.
The router you placed in the network will get an IP address and probably NAT the traffic of the client connected to you wireless router.
Another method to detect NAT devices on the network http://www.sflow.org/detectNAT/. The downside is that you need to have an sFlow collector for detecting. Also you want to automate disabling ports when a rogue NAT device has been detected. Implementing network access control is probably a more easy and reliable direction.