As you said, it's Typescript that returns the error message so I would say that the first step towards improving the error messages formatting would be to start at the root which is TypeScript itself. If they would agree to improve the formatting in the way your package does then it would already be at least 50% success.

This would still not allow messages to have markdown formatting since the LSP spec doesn't support that but if the whitespace formatting would be handled by Typescript then this package would have to do a lot less itself.