The best way to achieve your goal, is by using the IE Tab extension for your default browser (Firefox / Chrome).
Instead of trying to set a default browser for a specific application on the Windows side - which is way more complicated, do it on the browser side.
What Is IE Tab?
IE Tab for Chrome (and IE Tab 2 for Firefox) is a browser extension that allows you to use Internet Explorer to display web pages in a tab. This is useful in many cases where a web page requires Internet Explorer features.
Features
Auto URLs - Allows you to specify which URLs should always be loaded with IE Tab.
As you can see, the reason IE Tab is very useful in your case, is because it has the Auto URLs feature.
Instead of settings a specific browser for a specific application, you can simply tell IE Tab to be activated whenever you launch a certain URL.
And even better - it supports Regular Expressions, so whether your report is created as an html file offline (locally) or online, it will be very easy to tell IE Tab to always open your reports - which lets you view them in an IE environment - inside your default browser.
For example, if you reports are online, you can add http://www.myonlinereports.com/* to the Auto URLs to automatically open in IE Tab every page of the www.myonlinereports.com website.
If you reports are offline, you can add something like r/file:///D:/My%20Documents/My%20Reports/.* (don't forget the r/ prefix) to the Auto URLs to automatically open in IE Tab every file inside the My Reports folder.
