I can Curl, but I can't Guzzle

I’m trying to do a simple Mailchimp subscription in my custom module. I didn’t want to use a contributed module, just because I have one existing form, and I want to also subscribe users on form submit.

That said, I can get this to work using curl.

    $merge_vars = array(
      'FNAME' => $form_state->getValue('fname'),
      'LNAME' => $form_state->getValue('lname'),
      'MERGE3' => $address,
      'MERGE5' => $form_state->getValue('phone'),
      'MERGE6' => $form_state->getValue('birthday'),

  $data = array(
    'apikey'        => $api_key,
    'email_address' => $email,
    'status'        => $status,
    'merge_fields'  => $merge_fields

  $options = array(
    'method' => 'PUT',
    'data' => json_encode($data),
    'headers' => array(
      'Content-Type' => 'application/json',
      'Authorization' => 'Basic '.base64_encode( 'user:'.$api_key ), // maybe need this? formatted differently?

  $base_url = 'https://' . substr($api_key,strpos($api_key,'-')+1) . '.api.mailchimp.com/3.0/lists/' . $list_id . '/members/' . md5(strtolower($data['email_address']));
  $mch_api = curl_init(); // initialize cURL connection
  curl_setopt($mch_api, CURLOPT_URL, $base_url);
  curl_setopt($mch_api, CURLOPT_HTTPHEADER, $headers);
  curl_setopt($mch_api, CURLOPT_USERAGENT, 'PHP-MCAPI/2.0');
  curl_setopt($mch_api, CURLOPT_RETURNTRANSFER, true); // return the API response
  curl_setopt($mch_api, CURLOPT_CUSTOMREQUEST, 'PUT'); // method PUT
  curl_setopt($mch_api, CURLOPT_TIMEOUT, 10);
  curl_setopt($mch_api, CURLOPT_POST, true);
  curl_setopt($mch_api, CURLOPT_SSL_VERIFYPEER, false);
  curl_setopt($mch_api, CURLOPT_POSTFIELDS, json_encode($data) ); // send data in json

  $result = curl_exec($mch_api);

But for some reason, I can’t figure out the right combination of code that does the same thing with Guzzle. Here’s what I have that doesn’t seem to actually trigger a subscription…

  $client = Drupal::httpClient();
  $response = $client->get($base_url, [
    'auth' => ['apikey', $api_key],
    'headers' => [
      'content-type' => 'application/json',
      'authorization' => 'Basic '.base64_encode( 'user:'.$api_key ),
    'form_params' => json_encode($data)
  $data = $response->getBody();
  return $data;

Anyone have a little time to help me out? I’ve never used Guzzle before, so perhaps there’s a fair amount of ignorance here.

Drupal version: 

Source: https://www.drupal.org/taxonomy/term/4/feed