I did try build tools as well actually, it also doesn't work. I think you're right that it just isn't possible to install into a headless container (I spent a long time on this before I realized this...). Personally I would find a container image with the tools super helpful!
This might be more of a docker question than appveyor specific, but is there any way for me to use msbuild from our host platform? I'm not sure how to go about mounting the host drive to the container since it is being setup and run by the appveyor agent.
I think this is what we need that isn't already there:
* .NET Framework 3.5
* SQL Server (I think I actually managed to install this to the container with powershell though)
* NPM (also installed this with powershell just fine)