Its your router causing the issue by your own volition because as you say the connection works direct to the modem but not through the router, ergo the router is at fault and the support stops there because the is no actual issue with the connection.
Call the router manufacturer or replace it as Virgin will not investigate a fault with a router not supplied by them.
Also 2nd line will not investigate a fault with your connection which is fine when your 3rd party router is removed either as it is outside of the Support scope.
We do offer support through the PC Help helpline for problems such as this
http://www.virginmedia.com/help/pchelp/