PUT e Delete não funcionam com o ASP.NET WebAPI e o banco de dados no Windows Azure

Estou trabalhando em um projeto ASP.NET WebAPI com operações básicas de CRUD. O projeto é executado localmente e possui um banco de dados de exemplo que fica dentro do Windows Azure.

Até agora, o Http GET e POST funciona bem, dando-me 200 e 201. Mas estou lutando com DELETE e POST. Mudei os manipuladores no Web.config, removi o WebDav, mas nada disso funcionou. Também ativar o CORS e todos os tipos de atributos como [AcceptVerbs] não funcionou.

Alguma idéia do que estou fazendo de errado?

Saída bruta do violinista:

HTTP/1.1 405 Method Not Allowed
Cache-Control: no-cache
Pragma: no-cache
Allow: GET
Content-Type: application/json; charset=utf-8
Expires: -1
Server: Microsoft-IIS/8.0
X-AspNet-Version: 4.0.30319
X-SourceFiles: =?UTF-8?B?QzpcVXNlcnNcTWFyY1xPbmVEcml2ZVxEb2t1bWVudGVcRmlcVnNQcm9qZWt0ZVxONTIwMTQwODI1XE41XE41XGFwaVxwcm9kdWN0XDEwODM=?=
X-Powered-By: ASP.NET
Date: Sun, 14 Sep 2014 15:00:43 GMT
Content-Length: 75

{"Message":"The requested resource does not support http method 'DELETE'."} 

Web.config:

<system.webServer>
    <validation validateIntegratedModeConfiguration="false" />
    <modules runAllManagedModulesForAllRequests="true">
      <remove name="WebDAVModule" />
    </modules>
    <handlers>
      <remove name="WebDAV" />
      <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
      <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT" type="System.Web.Handlers.TransferRequestHandler" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0" />
    </handlers>
 </system.webServer>

Controlador:

 public class ProductController : BaseApiController
    {
        public ProductController(IRepository<Product> repo)
            : base(repo)
        {

        }

        [HttpGet]
        public IEnumerable<Product> Get()
        {
            //...
        }

        [HttpGet]
        public Product Get(int id)
        {
            //...
        }

        [HttpPost]
        public HttpResponseMessage Post([FromBody] Product product)
        {
           //...
        }

        [HttpPut]
        public HttpResponseMessage Put(int productId, [FromBody] Product product)
        {
            //..
        }

        [HttpDelete]
        public HttpResponseMessage Delete(int productId)
        {
            //..
        }

    }

Roteamento e formatadores:

 public static void Register(HttpConfiguration config)
    {
        // Web API configuration and services
        // Configure Web API to use only bearer token authentication.
        config.SuppressDefaultHostAuthentication();
        config.Filters.Add(new HostAuthenticationFilter(OAuthDefaults.AuthenticationType));


        config.Routes.MapHttpRoute(
            name: "Product",
            routeTemplate: "api/product/{id}",
            defaults: new {controller = "product",  id = RouteParameter.Optional }
        );

        // Custom Formatters:
        config.Formatters.XmlFormatter.SupportedMediaTypes.Remove(
            config.Formatters.XmlFormatter.SupportedMediaTypes.FirstOrDefault(t => t.MediaType == "application/xml"));

        var jsonFormatter = config.Formatters.OfType<JsonMediaTypeFormatter>().First();
        jsonFormatter.SerializerSettings.ContractResolver = new CamelCasePropertyNamesContractResolver();
    }
}

questionAnswers(3)

yourAnswerToTheQuestion