There should be a simple fix for the specific IE version issue. It's possible to run multiple IE browser engines on the same OS as there's an app that does this already (IETester).
It should be possible to create a piece of software - if one doesn't already exist - a web browser that is designed for enterprise use and allows the IT department to specify different browser engines for different sites.
That outdated internal application could be configured to use the IE6 engine - other newer applications and external websites could use newer IE engines or another engine entirely such as WebKit.
It could be rolled out by replacing 'iexplore.exe' and assuming the interface was very close to Internet Explorer it could even be transparent to users and thus very easy to roll out.