Unfortunately, you cannot change/extend the C# compiler to achieve what you want here. However, there is a Microsoft project called Roslyn which exposes a public API for implementing your own extensions to the C# compiling pipeline.
Using Roslyn's structures, it should be easy to traverse your source code syntax trees looking for methods that are not public/static. Once you find them, you could generate a code issue reporting the problem; those issues are shown both in the code editor (wavy underline) and in the Errors List panel. From Roslyn's official documentation:
The code issue provider makes it easy to surface an error or suggestion to the user as a wavy underline in the editor or appear in the Error List window.
Look around for examples of CodeIssueProviders; it could be useful for what you need to do.