Programmatic Impersonation Delegation For Remote Resources (Double-Hop)

I am having a hard time to get Runtime Impersonation to work.

Scenario:

Anonymous access is disabled in all servers, and windows auth is enabled
Client calls Web Api 1
Web Api 1 may call Web Api 2, or the oData Service
Call from Web Api 1 to Web Api 2 needs to be impersonated with the Client Credentials
Calls from Web Api 1 to oData Service must not be impersonated
Web Api 1 calls both the service using Web Request
We have Kerberos delegation configured properly

What Works (Kinda):

If I turn on impersonation in Web Api 1 using the Web.config

<authentication mode="Windows"/>
<identity impersonate="true"/>

All calls are getting impersonated. Web Api 1 --> Web Api 2, and Web Api 1 --> oData Service

This is not what we want. We want the oData Service to be accessible only via the Application Pool account. Hence we don't want to impersonate all outgoing calls from Web Api 1.

Programmatic Impersonation

We tried to impersonate only calls going from Web Api 1 to Web Api 2 using the following code

Disable Impersonation in Web.config

<authentication mode="Windows"/>
<identity impersonate="false"/>

Impersonate calls from Web Api 1 to Web Api 2.

// Impersonate the currently authenticated User
using (((WindowsIdentity)HttpContext.Current.User.Identity).Impersonate()) {
    var request = (HttpWebRequest)HttpWebRequest.Create(uri);
    ...
    ...
    request.Credentials = CredentialCache.DefaultCredentials;

    response = (HttpWebResponse)request.GetResponse();
};

Results:

Calls From Web Api 1 to oData are not impersonated (As expected)
Calls from Web Api 1 to Web api 2 are not impersonated either. This is the problem.

Question: