I was hoping for more information, but seems like your going for the less is more approach. I will do my best.
1. Ping the website when router is bypassed and find out what one of the IP is.
2. Hook the router back up, and make sure your online.
3. Ping the IP of the site.
If you cant ping the domain, but you can ping the IP, sounds like a DNS issue. Might want to try turning off Advanced DNS in 192.168.0.1>Setup>Internet>Manual Internet Connection Setup>Uncheck "ADVANCED DNS SERVICE".
Now if you can ping host/ip when direct, and can still ping host/ip through router, but cant get to the website, try this.
1. From CMD
2. Tracert <insert IP here>
See if there is any drops in the chain. A drop between two hops could show where TCP data is being blocked but ICMP data is going through. Unlikly but I don't know what your entire network consist of so...
Now if is a uninterupted trace from you to website with router and can ping both IP and domain with router in place, then it could be your firewall, but I find that extremly unlikly. Also, it could be something odd like the MTU or something. Whats the protocol for the site? FTP? HTTP? HTTPS?