I have lots of HttpClient dependency-injected services. I use an HttpClientFactory to get these.

There are variations, but in a simple case:

// Configure a named and configured HTTP client called "MyHttpClient"
var httpClientBuilder = services.AddHttpClient("MyHttpClient", (sp, client) => /* configure the client */);

...

// Set up the DI service with a factory that gets the named "MyHttpClient"
services.AddTransient<ICustomService>(
    s => 
        new CustomService(
            s.GetRequiredService<IHttpClientFactory>().CreateClient("MyHttpClient")));

I have a service that responds with Set-Cookie headers (in this case AWSALB load balancing indicators from AWS, but this question is for any cookies) – I want the HttpClient to respond with these cookies.

I can take that httpClientBuilder and use it to create a custom handler that sets up a CookieContainer to do this, but it feels like re-inventing the wheel.

Is there an existing solution for this? Something like:

services.AddHttpClient(...)
    .AddCookieHandler(); // Automatically set up `CookieContainer` for all requests