The request message was already sent. Cannot send the same request message multiple times

You are calling the same func parameter twice:

var response = await ProcessRequestAsync(func);
//...
response = await ProcessRequestAsync(func);

In this case func returns the same request every single time. It doesn’t generate a new one every time you call it. If you truly need a different request each time then func needs to return a new message each call:

var response = await GetAsync(() => new HttpRequestMessage()); // Create a real request.

public async Task<HttpResponseMessage> GetAsync(Func<HttpRequestMessage> requestGenerator)
{
     return await RequestAsync(() => requestGenerator());
}