I can successfully make the RESTful post using cURL:

$ch = curl_init();

curl_setopt($ch, CURLOPT_URL, 'https://api.mindbodyonline.com/public/v6/usertoken/issue');
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, "{n    "Username": "Siteowner",n    "Password": "apitest1234"n}");

$headers = array();
$headers[] = 'Content-Type: application/json';
$headers[] = 'Api-Key: {myapikey}';
$headers[] = 'Siteid: 999999';
curl_setopt($ch, CURLOPT_HTTPHEADER, $headers);

$response = curl_exec($ch);
print_r($response);
if (curl_errno($ch)) {
    echo 'Error:' . curl_error($ch);
}
curl_close($ch);

However, what seems to be the same request using wp_remote_post, fails to deliver the “Api-Key” in the headers:

$headers = array();
$headers[] = 'Content-Type: application/json';
$headers[] = 'Api-Key: {myapikey}';
$headers[] = 'Siteid: 999999';

$response = wp_remote_post( 'https://api.mindbodyonline.com/public/v6/usertoken/issue', array(
    'method' => 'POST',
    'timeout' => 45,
    'redirection' => 5,
    'httpversion' => '1.0',
    'blocking' => true,
    'headers' => $headers,
    'body' => array( 'Username' => 'Siteowner', 'Password' => 'apitest1234' ),
    'cookies' => array()
    )
);

if ( is_wp_error( $response ) ) {
   $error_message = $response->get_error_message();
   return "Something went wrong: " . $error_message;
} else {
   echo 'Response: <pre>';
   print_r( $response );
   echo '</pre>';
   return;
}

Does anyone spot an obvious mistake in my syntax, or have a recommendation on where to troubleshoot? I see that wp_remote_post wraps the WP_Http class.

Update:

Confirmed that the $headers argument being sent to Requests::request looks like this:

Array
    (
        [0] => Content-Type: application/json
        [1] => Api-Key: {myapikey}
        [2] => Siteid: 99999
    )

Inspecting into the Requests::request method, it appears that the $transport (in this environment) is the following object:

Requests_Transport_cURL Object
(
    [headers] => 
    [response_data] => 
    [info] => 
    [version] => 468480
    [handle:protected] => Resource id #4
    [hooks:protected] => 
    [done_headers:protected] => 
    [stream_handle:protected] => 
    [response_bytes:protected] => 
    [response_byte_limit:protected] => 
)

And within it’s request method, header array is still as above.

Update two:

This is interesting. In Requests_Transport_cURLs setup_handle method, when the CURLOPT_HTTPHEADER is set, $headers looks like this:

Array
(
    [0] => 0: Content-Type: application/json
    [1] => 1: Api-Key: a3f5be6229744000b9bc25f603e80c45
    [2] => 2: Siteid: -99
    [3] => Connection: close
)

On line 385:

385    curl_setopt($this->handle, CURLOPT_HTTPHEADER, $headers);

So perhaps I need to be sending a hash of key-value pairs as opposed to an array of json strings.