I got the problem solved, with the help of L3 support from DLink.
The tech used Teamviewer to connect, but the router's reset cycle kicked him out. He concluded my setup was double nat'ed, and since I couldn't access my modem, he concluded the modem had a DHCP server set up. The solution was to disable NAT on the router and use it as a switch.
That works, and we ended the support session, but there are a few problems:
1) My modem doesn't have a DHCP server. It's a TPLink TD-8616: http://www.tp-link.com/us/products/details/?categoryid=221&model=TD-8616#spec
2) I couldn't connect to my modem because it's in transparent bridge mode, and the only way to access it is to reset it.
3) The DSL service from the ISP (Sonic.net) is DHCP. By turning off DHCP on the router, all my devices are routed by Sonic.net's router. I'm only paying for one IP address, so the ISP will likely eventually question my setup.
Sonic.net offers one static IP address for no additional cost, so I switched my service to StaticIP. After setting up the router with that info, it looks like the connection is working AND I can use the DHCP server on the modem.
Thanks for your suggestions, though. Looks like you don't need to connect in remotely.