I cant comment (not enough rep on this site) so I have to submit this as an answer.
1) I have successfully set up a SharePoint SSRS portal using forms authentication (FBA). The SSRS part was not hard - very straightforward actually, just perform the SSRS install as SharePoint integrated. However, the SharePoint FBA part was very painful (not difficult, just lost a lot of hair and sleep over little stuff you didnt think would be necessary and couldnt find straight answers to.)
I know that you can use Report Builder 3.0 to save to a SharePoint site. I have not tried that using the FBA SharePoint portal. I dont work at that company anymore, so I dont have access to test that either. I would think it is very possible though. The portal I set up was accessed by non-domain users, which is why I had to use FBA. I was using SharePoint 2007 and SQL Server 2008 R2.
I know the following link tells of the permissions needed for Report Builder to save to a SharePoint library: http://technet.microsoft.com/en-us/library/dd255209(v=sql.105).aspx
2) I did find a specific blog post about using Basic Authentication: http://blogs.msdn.com/b/bobmeyers/archive/2006/09/28/how-to-enable-report-builder-for-non-domain-users.aspx.
There is also an article on MS technet titled: "How to: Configure Report Builder Access"
3) Another thing that is worth trying - You can use a domain account for report viewing only, then use the ReportViewer Web service/API/webpart/etc to have a report embedded within a web application. That way, the report connects using the username/password specified, not the user viewing the application. That has worked for an application I helped work on that was exposed on a DMZ web server, but used an SSRS report that was on the internal network. In that case, the SSRS instance was a Native install. If you are using .NET - you can use Visual Studio. It already has controls built in.
The only think I dont know about in this case is the use of Report Builder. Just thought I would mention it because it might help spur some thoughts.