ServiceReferences.ClientConfig use basicHttpsBinding or similar

Question

We're trying to migrate an IIS-hosted Silverlight application using WCF from HTTP over to using HTTPS.

We have the following problem:

When loading the application, we can see the SL loading spinner with blue dots that just keeps at being on "100%".

When debugging, I can see that in this specific config, basicHttpsBinding is not an allowed attribute, although being valid in other configs.

When trying to use customBinding, there seems to be a security tag mandatory. Adding it gives out the error that the authenticationMode attribute is missing. When looking up the attribute for that tag for the customBinding, there's a Microsoft article on it. However, even trying out the supposedly "default" value sspiNegotiated on the "optional" tag only yields that that value isn't known.

Stepping into the code to find more info about possibly allowed values does not seem to be possible.

This is the ServiceReferences.ClientConfig for the client that uses one of those services. The services also seem to be set up to use HTTPS already.

<configuration>
  <system.serviceModel>
    <bindings>
      <basicHttpsBinding>
        <binding name="config" 
        maxBufferSize="2147483647"
                receiveTimeout="01:00:00" 
        sendTimeout="01:00:00" 
        closeTimeout="01:00:00" 
        openTimeout="01:00:00"
                maxReceivedMessageSize="2147483647">
            <security mode="Transport">
                <transport clientCredentialType="Windows" />
            </security>
            <httpTransport />
        </binding>
      </basicHttpsBinding>
    </bindings>
    <client>
      <endpoint address="https://localhost/namespace.service/SomethingService.svc"
          binding="basicHttpsBinding"
          bindingConfiguration="config"
          contract="namespace.proxy.ISomethingService"
          name="BasicHttpBinding_something" />
    </client>
  </system.serviceModel>
</configuration>