The answer is so simple that I'm amazed I didn't think of this before, and even more amazed that nobody else seemed to think of it before either (or at least didn't write about it).
You will need Robert McLaws's MSBuild Toolkit, which you can find here. The toolkit enables you to compile Visual Studio 2005 projects for .Net 1.1. The catch is that it seems to be restricted to class libraries and Windows Forms applications (though I didn't test the latter). Since the company where I work specializes in Web Applications, we were left with a problem: what to do with our Web apps?
Now think of this: to Visual Studio 2003, web applications are really nothing more than class libraries that happen to contain a few aspx files. The only difference is in the .csproj files, where we have VisualStudioProject/CSHARP[ProjectType = "Web"] for web applications, and VisualStudioProject/CSHARP[ProjectType = "Local"] for class libraries. So I took one of our Asp.Net 1.1 Web Applications, edited the .csproj file and wrote Local instead of Web. I then opened the project in VS2005, and compiled it for .Net 1.1.
I opened the web application in IE, and to my utter satisfaction it actually worked!
More amazingly, even the design view of the aspx files works. The connection between the aspx files and the codebehind files is lost, though. This means that when you add an asp.net control to the aspx file, you have to manually declare the corresponding protected field in the aspx.cs file. This is a trade-of that I'm willing to live with: losing this connection allows me to use Visual Studio 2005 and, more importantly, Team System.