It's amazing that this was left unanswered Gavin. You would think a notification styled error handling would be a default action for API calls. I did email the Formidable team about this question and they supplied a partial answer.
They referenced this hook: https://formidableforms.com/knowledgebase/frm_api_post_response/
I attempted to utilize it to fire a wp_mail() function on the else condition, but the hook does not appear to fire when in my functions.php. I'm wondering if you had been able to solve your original inquiry from 5 months ago?
Thanks, Steve

Hey Steven. I found the same KB article and managed to use the examples to update a field in the record with the API status response.

add_action( 'frmapi_post_response', 'frm_save_api_response', 10, 3 );
function frm_save_api_response( $response, $entry, $form_action ) {
  $status = wp_remote_retrieve_response_code( $response );
    FrmProEntryMeta::update_single_field( array(
      'entry_id' => $entry->id,
      'field_id' => 11,
      'value'    => $status,
    ) );
}

Then, when field_id is anything other then a '200' I give the user an error and also send a Slack notification to our support team to investigate further.
This has been working well so far.
Regards,
Gavin

Hey Steven. I found the same KB article and managed to use the examples to update a field in the record with the API status response.

add_action( 'frmapi_post_response', 'frm_save_api_response', 10, 3 );
function frm_save_api_response( $response, $entry, $form_action ) {
  $status = wp_remote_retrieve_response_code( $response );
    FrmProEntryMeta::update_single_field( array(
      'entry_id' => $entry->id,
      'field_id' => 11,
      'value'    => $status,
    ) );
}

Then, when field_id is anything other then a '200' I give the user an error and also send a Slack notification to our support team to investigate further.
This has been working well so far.
Regards,
Gavin