Using Process Monitor, I couldn't find anything definitive going on in there in terms of errors, though I was surprised that it seemed like the browser.exe process was accessing practically all of the temporary internet files on the machine, as well as the cookies index.dat and history files (showed as the same PID for all browser.exe entries).
For giggles, I tried copying the install files to their Exchange server and running the install - I ran into the same problem there. On the original server, I had added the machine name, private IP, and loopback IP into trusted sites in IE security zones, but on the Exchange box I decided to go a bit further. I added the trusted sites on that server appropriately, and then switched Windows Browser Applications from Disable to Enable on all zones. When I tried to run the installer again, I got the message "Windows cannot access the specified device, path, or file. You may not have the appropriate permissions to access the item."
I've switched directories, recopied the whole thing into a new directory, granted domain\everyone full control to the directory, and undid my IE setting changes above, and still get the same error over there. I can create and edit other files in the directory just fine.
One thing that's interesting to note is I went to windowsupdate.microsoft.com on the Exchange server, and it prompted that a pending patch install required a reboot. Came to find out it was IE7, or at least IE7 was part of the last update done through our managed services software. After the reboot, I still got the error trying to run browser.exe above, but I will reboot the main server tonight and see if the pending IE7 install is part of the problem.