Not very smart with networking, if I run into a problem I use lots of googling until I get it running. I also may use incorrect terms, sorry.
It sounds like what I ran into with one of my remote sites I set up. Router/modem at one end of the basement to feed the cable TV and phone circuits. I pulled a long run of cat 5 into a router with better wifi coverage and where my thin client running ProScan & scanner are located.
I was going crazy with trying to get the feed to leave the house. Finally, I plugged into the first device in line, and edited that device to allow all the ports to be controlled by the second one .
My suggestion would be to unplug the network line that comes from outside. Plug that into the computer and try to access internet. If you can, I would guess there may be some settings in that outside device that can be changed. Try to edit the device settings with 10.0.0.1 or 192.168.1.1 (or 0.1). If you are successful, you can then try to have it allow all ports to be controlled by the second router.
On my thin client, I have a No-IP.com Dynamic DNS Update Client that checks the IP address every couple minutes and then updates the source address of my no-ip addresses assigned to my feeds.
chris