A bit of a ramble I know ..........
The case summary is 3 pages (sides) long.
The evidence list says "B. Case Summary & Traffic Management Order" and "Total no. of sides = 3". That suggests to me that the TMO has not been included. If it had been included then the number of sides would be larger.
The problem thus far has been that all you have is strong mitigation. Neither the Council or the Police instructed you to park in the pay & display bay, albeit temporarily, whilst you went to find somebody to provide guidance. So at first sight the PCN was lawful and correctly issued and the Council is entitled to pursue it.
However the Adjudicator cannot rule on mitigation, only the facts, the balance of probability and whether or not correct procedure has been followed.
The Adjudicator can (rarely done) ask the Council to reconsider and cancel which is probably what you were aiming at.
In your case the Notice of Rejection seems to be your only avenue of procedural defect. The Council are obliged to consider your formal representations against the Notice to Owner. Your reps were clear and comprehensive however the NoR is very sparse with minimal detail suggesting a boilerplate rejection with little or no actual consideration. Conversely the subsequent case summary is much more comprehensive. Seems they only really considered your representations when you submitted an appeal to the Adjudicator.
However the Council is also obliged to submit evidence to the Tribunal as to the contravention and why they believe it occurred. The Traffic Order creates the alleged contravention and should be presented in evidence. If the Traffic Order is really not there then you can also argue that there is no evidence to lawfully establish the alleged contravention.
Also Section C says the penalty outstanding is £130. I would point it out to the Adjudicator as an unlawful demand for money. The figure should be £80. The Adjudicator might well dismiss that as just a typo and not relevant.