As I understand your case, you're saying that a blue badge holder would be exempt if they applied for a permit, but on this occasion a permit had not been applied for.
Therefore while a case can be made that there is no traffic management purpose in enforcing this PCN, by the letter of the law a contravention did occur. Unfortunately therefore this is a case of mere mitigation, and the tribunal cannot allow an appeal based on mitigating circumstances.
So you'd be reliant on two possible ways out of this:
1) The adjudicator finds that the authority did not properly consider the representations made, which allows the tribunal to accept a collateral challenge and allow the appeal, or
2) The adjudicator invites the authority to reconsider the mitigation provided.
The issue with point 1 is that it is very much down to the adjudicator on the day, some adjudicators take the view that if the notice of rejection says that the representations have been considered, they can't go behind that.
The issue with point 2 is that even if the adjudicator accepts that the mitigation in this case is compelling enough to warrant a reconsideration, there is no statutory power direct a reconsideration in a moving traffic case, so the adjudicator cannot force them to reconsider. Therefore if they simply ignore the adjudicator, the adjudicator would be compelled to refuse the appeal.
It is true that Croydon is a bit rubbish at putting its evidence packs together in the first place, and it is always possible they might mess up the evidence pack, but even taking that into account I would judge your chances of winning at 50/50 at best.
Of course this assumes the driver lives inside the relevant zone to begin with, if this was just a visitor from outside of the area who just happened to have a blue badge, the exemption isn't applicable in any event.