My biggest confusion is that the SME server does not see the router.
What do you mean by "does not see"?
Therefore, adding a local network does not work.
Again, please define "does not work".
What you need is for the router's "WAN" address to be on the SME server's local network. Add the router's local network as an additional local network, using the router's "WAN" address as the router to that network. Done.
> The server, however, can't ping anything that isn't on the original network.
This will be because the "router" is operating as a firewall, using NAT. If you can configure it as a pure router, ping will be able to traverse the router from WAN port to LAN.
Be sure to lock down wireless access to your router, otherwise passers by will be able to do evil things, such as send spam via your server.