in API, create multiple controller constructor with one parameter

When using dependency injection, you should only have one constructor where all dependencies can be satisfied. Otherwise, how is the DI container to know which constructor to utilize? That's your issue here. Using the Microsoft.Extensions.DependencyInjection package, and since this is a controller you're injecting into, there's only one reasonable way to solve this: don't register one or the other of the services, IDigitalDocumentService or IDatatabaseInitializer. If only one is registered, the service collection will simply use the constructor it has a registered service for.

It's possible with a more featured DI container, you might be able to configure something to allow it choose the proper constructor. How to do that would be entirely dependent on the DI container you end up going with, though, so not much more can be said on the subject at this point. Just realize that the default container (Microsoft.Extensions.DependencyInjection) is intentionally simplistic, so if you needs are more complex, you should sub in a full DI container.

UPDATE

You should be doing integration testing with the test host and an in-memory database. The basic approach is:

public MyTests()
{
    _server = new TestServer(new WebHostBuilder().UseStartup<TestStartup>());
    _context = _server.Host.Services.GetRequiredService<MyContext>();
    _client = _server.CreateClient();
 }

In your app's Startup, create a virtual method:

public virtual void ConfigureDatabase(IServiceCollection services)
{
    // normal database setup here, e.g.
    services.AddDbContext<MyContext>(o =>
        o.UseSqlServer(Configuration.GetConnectionString("Foo")));
}

Then, in ConfigureServices, replace your database setup with a call to this method.

Finally, in your test project, create a TestStartup class and override the ConfigureDatabase method:

public class TestStartup : Startup
{
    public override void ConfigureDatabase(IServiceCollection services)
    {
        var databaseName = Guid.NewGuid().ToString();
        services.AddDbContext<MyContext>(o =>
            o.UseInMemoryDatabase(databaseName));
    }
}

Now, in your tests you just make requests against the test client (which is just an HttpClient instance, so it works like any other HttpClient). You start by setting up your database with appropriate test data, and then ensure that the correct response is returned:

// Arrange
_context.Add(new DigitalDocument { Name = "Foo" });
await _context.SaveChanges();

// Act
// Submit a `DigitalDocument` with the same name via `_client`

// Assert
// Inspect the response body for some indication that it was considered invalid. Or you could simply assert that no new `DigitalDocument` was created by querying `_context` (or both)

This is admittedly a lot easier with an API, as with a web application, you're going to invariably need to do some HTML parsing. However, the docs and corresponding sample app help you with that.

Additionally, in actual practice, you'd want to use a test fixture to prevent having to bootstrap a test server for every test. Again, the docs have you covered there. One thing to note, though, is that once you switch to using a fixture, your database will then be persisted between tests. To segregate your test data, make sure that you call EnsureDeleted() on your context before each test. This can be easily done in the test class' constructor:

public class MyTests : IClassFixture<WebApplicationFactory<Startup>>
{
    private readonly HttpClient _client;
    private readonly MyContext _context;

    public MyTests(WebApplicationFactory<Startup> factory)
    {
        factory = factory.WithWebHostBuilder(builder => builder.UseStartup<TestStartup>());
        _client = factory.CreateClient();
        _context = factory.Server.Host.Services.GetRequiredService<MyContext>();
        _context.EnsureDeleted();
     }

I don't even like this much bootstrapping code in my tests, though, so I usually inherit from a fixture class instead:

public class TestServerFixture : IClassFixture<WebApplicationFactory<Startup>>
{
    protected readonly HttpClient _client;
    protected readonly MyContext _context;

    public TestServerFixture(WebApplicationFactory<Startup> factory)
    {
        factory = factory.WithWebHostBuilder(builder => builder.UseStartup<TestStartup>());
        _client = factory.CreateClient();
        _context = factory.Server.Host.Services.GetRequiredService<MyContext>();
        _context.EnsureDeleted();
     }
}

Then, for each test class:

 public class MyTests : TestServerFixture
 {
     public MyTests(WebApplicationFactory<Startup> factory)
         : base(factory)
     {
     }

This may seem like a lot, but most of it is one-time setup. Then, your tests will be much more accurate, more robust, and even easier in many ways.