To answer my own question, and in part because I didn't like the way that QUrlOperator works in the first place, I've put together a quick hack that lets you do a HTTP get as a static method. Ie:
#include "tturloperator.h"
QString page = TTUrlOperator::getPage("http://www.cnn.com");
It will retrieve urls of type 'http' or 'file'. I didn't have a chance to add FTP. This just uses raw sockets, by the way. It doesn't do anything fancy with headers nor have I added in proxy code (although that should be a snap).
There is other code in there to replicate QUrlOperator but I've commented it out to reduce executable size. If anyone wants anything added let me know and I'll try to put something together.
You can get it
here.