We're at the notice to owner stage already so there are no informal reps, only formal ones.
These can be kept fairly simple:
Dear Lincolnshire County Council,
The alleged contravention did not occur, because a valid permit was on display, I attach a zoomed-in photo taken by the CEO, and also a seperate photo showing the permit. This is the first PCN received and no other PCN has been received before or since, which is to be expected as a valid permit has been obtained for this vehicle.
Furthermore while I appreciate the PCN was validly served, I did not find it on the car, therefore I lost the opportunity to pay the penalty at the discounted rate.
In the circumstances, please would you cancel the PCN.
Yours faithfully,
Send the representations online and keep a screenshot of the confirmation page.