If your DS Lite can't connect to your router, it's more than likely using a WPA connection. I use a WPA connection myself, all that means is that you'll only be able to use your 3DS to connect to Pokecheck. No need to fiddle with your router more than likely.
(I'm not sure if I was clear enough conveying this in my last message, so I'll try rephrasing it.)
When you come to the 3DS equivalent of the middle screen, make sure that you only have one connection listed, the one with the DNS set to connect to Pokecheck. If you have an option to connect normally, your DS most of the time will. This is my best guess as to where you're going wrong.
If my attempt at trouble shooting didn't help, then I might know people that can. Pokecheck has a tech-support-esc chat in the form of the shoutbox. The users there can be a huge help at narrowing down your problem. Tell them that Mantyke sent you and I'm sure they'll be happy to help.