You're on the right track. You're probably losing connection on the WDTV because your home router is not allowing the Slingbox to stream in TCP (Top Connection Protocol). During Slingbox setup, most of the time the router configures itself to allow TCP streaming for the Slingbox, but a handful of them do not. This results in the Slingbox streaming in Relay, which is known to cause connection issues when using connected devices like the WDT.V That's probably the case here. When this happens, port forwarding needs to be manually set in the router at home.
Here is a quick article, which contains a link to help you through the port forwarding process in your home router:
You will want to set up port forwarding for port 5201 for the Slingbox.
Hope this helps!