I wasn't aware that you could only have one browser installed on a computer at a time. What's wrong with installing Firefox for 99% of tasks, and also having IE6 available for the obsolete and soon to be extinct tasks that require it?
What's wrong is that it costs time and money for the variety of things that go in to supporting and maintaining an additional application. The bean counters would throw a fit at the idea of spending money on two applications that do effectively the same thing.
Could you explain how it would cost a significant amount of time/money just to deploy Firefox across the machines? If you have proper app-deployment infrastructure in place, it shouldn't take more than a few clicks. Then send out a note saying "you can use Firefox for all the non-intranet stuff, just remember that we don't officially support it". And you're all set. Firefox auto-updates itself so that's taken care of as well. Eventually one could start using IE Tab+Firefox and remove IE (the icon). IE Tab can automatically open pre-defined websites in an IE instance within Firefox itself, thus the end user only has to deal with one browser (icon). When they visit the IE6-only site, it opens up in IE tab, but all other sites would work in Firefox. Problem solved.