What if you write your own little application requiring admin rights and bypass the MSI GUI altogether by launching your setup silently? The application can then read all the web site stuff and use a list box to select the right one? There might even be sample code for this in WIX that I am not aware of.
You should be able to launch the setup silently via a standard msiexec.exe command line:
msiexec.exe /I "C:\Test.msi" /QN /L* "C:\msilog.log" SITE=http://www.example.com/
Or use DTF included with the WIX toolkit. See some info here: https://serverfault.com/a/596519/20599
One thing to be aware of is that the listing of the web pages may not be related to admin rights as much as privileges. A custom configuration could mean that even generic admin rights wouldn't work - but I guess that is rare.