I finally figured out that Visual Studio keeps track of how you create a project (in other words which project template you select initially) and filters your options later based on that initial decision. The information is kept in the *.csproj file as a <ProjectTypeGuids> element.
Other than just editing the *.csproj file, is there a "right" way to change a project type for an existing project?
Considering the significance of that setting it seems likely there's a place in the GUI to change it, but I couldn't find one. Thanks!