Interpolated strings are simply converted to calls to string.Format. So your above line actually reads
private const string WEB_API_PROJECT = string.Format("{0}project.json", WEB_API_ROOT);
And this is not compile time constant as a method call is included.
On the other hand, string concatenation (of simple, constant string literals) can be done by the compiler, so this will work:
private const string WEB_API_ROOT = "/private/WebApi/";
private const string WEB_API_PROJECT = WEB_API_ROOT + "project.json";
or switch from const to static readonly:
private static readonly string WEB_API_PROJECT = $"{WEB_API_ROOT}project.json";
so the string is initialized (and string.Format called) at the first access to any member of the declaring type.