How to automatically evaluate comment content without user seeing the delay?

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The desired results mentioned in the question were achieved by using the wp_schedule_event() function, now:

  1. The user always sees the “awaiting moderation” message without experiencing any delay.
  2. The API requests happen without the users experiencing delay, they happen asynchronously when anyone visits the page and the time interval given as argument to the wp_schedule_event() has passed.

Notice that the wp_schedule_event() line should only run once since the plugin gets activated, one way to do this is:

if ( ! wp_next_scheduled( 'bl_cron_hook' ) ) {
  wp_schedule_event( time(), 'five_seconds', 'bl_cron_hook' );
}

You can read more here.