Solved!!!
I dug into the ACF core for a while and after a ton of trial and error, I found a super simple way to do this.
I created a custom endpoint that does the following:
// Get new post's ID
$post_id = $request['id'];
// Get new post's meta fields
$meta_fields = get_post_meta($post_id);
// Save the meta fields to the post
acf_save_post($post_id, $meta_fields);