Does your router not have a DMZ option?
To put it simply, If you set your switch to operate in the DMZ (Demilitarized Zone) your router puts it outside the NAT. If you tell me your router make/model I could possibly help you. Most routers have this option.
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Or you can just forward ALL ports to your Switch through NAT.
http://en-americas-support.nintendo.com/app/answers/detail/a_id/651/~/how-to-set-up-port-forwarding
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Network Address Translation is a simple routing function (hence the name router) it does not dynamically change. What you are probably hearing about is UPnP or Universal Plug and Play in which, to put it simply, the router senses what ports you need to be open and dynamically opens them while you are playing the game.
Unfortunately, my router does not have any DMZ settings, and whenever I try to forward all ports to the Switch, the same error occurs.
The main reason I though that NATs change is that I've read about how some people do several Internet Connection tests on the Switch, and the tests gives results which vary between NAT type A and NAT type B at different types.