Well, first off, what they are most likely testing for is wether the ipad has been opened/tampered with or not, because if it has they will probably say that the charging issue is caused by tampering. I assume what they mean by a "paper" they mean a piece of magnetic sensitive viewing film, this is a film that is sensitive to magnetic fields and when placed over something it will show the outlines of magnetic parts. How this can be used to determine if the ipad has been opened? I don't really know. but most likely they have magnets positioned in ways that they can tell if the case was opened after it left the factory.
I dont know why they wouldn't be able to do it til they had one in stock, but probably it is because they want to make sure they have one to send back to you as soon as possible.
If you are interested, there are several cool videos on youtube that explain magnetic viewing films
Sorry I am not an expert on any of this, just making educated guesses, but i hope this is helpful!