That would be to assume that a "background check" can be made. I am not aware of such a procedure being available (or, admittedly, not available).
If such a facility
does exist, it seems rather prejudicial to an individual to refuse to employ him based on something he hasn't actually done (a third missed test) and up to which point he has not in fact done anything sanctionable. We are not, after all, obliged to declare a police caution in job applications.
So unless he is being sanctioned, that individual's "record", if such a thing exists, ought to remain private. It should be enough that, if he chooses to hide what he could himself have volunteered to his employer and he then screws up again, the consequences - i.e. instant dismissal - are clear enough to both parties.
EDIT:
JV seems to confirm that this info is not available