how can you test the 'catch all route' using ordinary controller tests?

Question

Note: As per RafaeldeF.Ferreira's suggestion, this question has been heavily edited since its original form.

My JSON-based app needs to return something sensible when given a bad route. We already know that the following rescue_from ActionController::RoutingError doesn't work in Rails 3.1 and 3.2:

# file: app/controllers/application_controller.rb
class ApplicationController < ActionController::Base
  protect_from_forgery
  rescue_from ActionController::RoutingError, :with => :not_found
  ...
end

(This is well documented in https://github.com/rails/rails/issues/671.) So I implemented what José Valim describes in this blog entry (item 3), and details are provided below.

But testing it has been problematic. This controller rspec test:

# file: spec/controllers/errors_controller.rb
require 'spec_helper'
require 'status_codes'
describe ErrorsController do
  it "returns not_found status" do
    get :not_found
    response.should be(StatusCodes::NOT_FOUND)
  end
end

fails with:

ActionController::RoutingError: No route matches {:format=>"json", :controller=>"sites", :action=>"update"}

Yet this integration test calls ErrorsController#not_found and succeeds:

# file: spec/requests/errors_spec.rb
require 'spec_helper'
require 'status_codes'

describe 'errors service' do
  before(:each) do
    @client = FactoryGirl.create(:client)
  end

  it "should catch routing error and return not_found" do
    get "/v1/clients/login.json?id=#{@client.handle}&password=#{@client.password}"
    response.status.should be(StatusCodes::OK)
    post "/v1/sites/impossiblepaththatdoesnotexist"
    response.status.should be(StatusCodes::NOT_FOUND)
  end
end

So: Is there a way to test the 'catch all route' using ordinary controller tests?

implementation details

If you want to see the implementation, here are the relevant code snippets

# config/routes.rb
MyApp::Application.routes.draw do
  ... all other routes above here.
  root :to => "pages#home"
  match "/404", :to => "errors#not_found"
end

# config/application.rb
module MyApp
  class Application < Rails::Application
    config.exceptions_app = self.routes
    ...
  end
end

# config/environments/test.rb
MyApp::Application.configure do
  ...
  config.consider_all_requests_local       = false
  ...
end

# app/controllers/errors_controller.rb
class ErrorsController < ApplicationController
  def not_found
    render :json => {:errors => ["Resource not found"]}, :status => :not_found
  end
end