For systems connected to the proxy router, where the router uses the Fiddler system as the gateway, DNS will work if you enable IP forwarding on the gateway/Fiddler system following a reboot. Follow this registry change:
https://support.microsoft.com/en-us/kb/99686
As you surmised above, the trick is to get the gateway/Fiddler system to respond to traffic forwarded to it from the router.