搜查官方文档后发现：
HttpWebRequest.AllowAutoRedirect Property (System.Net) | Microsoft Learn

微软提供的http类库HttpClient （HttpWebRequest\WebClient已不推荐使用，用HttpClient代替）有备注提醒：当使用自动重定向时，认证头会自动清除，目的是保证安全问题。至此问题清晰，找类库代替。

解决方案：
使用 Flurl库

string url = "http://127.0.0.1:8080/test/api";


var response = url.WithHeader("Content-Type", "application/json")
    .WithHeader(
        "Authorization",
        "Basic"
    )
    .WithSettings(s => s.Redirects.Enabled = true)
    .WithSettings(s => s.Redirects.ForwardAuthorizationHeader = true)
    .WithSettings(s => s.Redirects.ForwardHeaders = true)
    .WithHeader("Expect", "100-continue")
    .PutStringAsync(File.ReadAllText(filePath))
    .GetAwaiter()
    .GetResult();
// Handle the response
Console.WriteLine(response.GetStringAsync().GetAwaiter().GetResult());