it sounds like a access control issue.
if you disabkle access controls...remove any web lists etc, does that workaround the issue ?
if so a custom firewall -> packet filtering -> custom filrewall rule
like this may assist
iptables -t nat -I ContFilt -d 22.214.171.124/32 -j RETURN
where 126.96.36.199 is the IP address of the web site you are trying to access.
Checking that right now. I also have 2 Internet connections in round robin, will disable one and see what happens. Will let you know.
Custom rule worked!