How to use the HTTP API (wp_remote_get) instead of cURL?

So it looks like I’ve managed to fix my own problem here. I don’t know if this is the best thinkable solution to this exact problem, but it does work!

My assumptions about CURLOPT_FOLLOWLOCATION seem correct. As Sally CJ also explained in the comments, the default setting in WP is always set to false. That’s why the data always contained contents from the (302) redirect instead of the actual targeted URL.

I’ve found this old thread and answer where the underlying CURLOPT_FOLLOWLOCATION is changed through the http_api_curl action. Using this action to override the default setting was the key for fixing my problem with wp_remote_get. With the function enabled (see below), my returned data is exactly the same as when using cURL.

function __set_curl_to_follow( &$handle ) {
    curl_setopt( $handle, CURLOPT_FOLLOWLOCATION, true );  
}
add_action( 'http_api_curl', '__set_curl_to_follow' );

Special thanks to Sally CJ and bosco for thinking with me on this!