Hook to trigger when a field collection item is being deleted
What you should do instead of an API call on delete is create a job with the QueueAPI in Drupal to create a job for processing later. In that job, pull the ID from the record (stored in the job) and do the API call. If the status failed, requeue the job. 

I can’t write all the code out for you but there are plenty of QueueAPI examples for D7 out there.

https://api.drupal.org/api/drupal/modules%21system%21system.queue.inc/group/queue/7.x

and

https://rbayliss.com/blog/drupal-queue-api/

Are great starting points.

You could then create a custom Drush command that processes  these jobs as fast as you desire. Note the section in rovs post that deals with “failed” states of a job and not removing it from the queue. That is key.

This will let Drupal function normally and defer errors in the remote system to you to work through.