Thank you Kenny, that actually helped, I understand the issue now.
They were right about using that function, it updates the fields and now the get_field() returns correctly. However the issue is not within this plugin either, as they mention that the reference is not created properly when the post is added to the database. If you use any custom AJAX plugin to add the posts (like they mentioned), that might have something missing there, so the references are not created.
I have added a bypass to the code to update the field before parsing, now it seems to be returning the correct field values in every case. However please note that I cannot add this fix to the live code within the upcoming release. I recommend leaving the plugin with the currecnt release, and not updating soon to make sure the modification remains.
The changes were made to the wp-content\plugins\ajax-search-pro\includes\classes\etc\class-asp_helpers.php file, I have made a copy of it here, in case it’s needed in the future.