IMO, although a driver is allowed a reasonable period to pay, it is your burden to prove. It's possible that you might convince an adjudicator just with your assertions and credibility, but as I read your posts the objective evidence doesn't support this.
By the same token, it's initially the council's burden to prove the contravention and as I understand it there are numerous adjudication decisions which find that the grounds of 'without payment of the parking charge' cannot be sustained when the council's policy is:
On-street parking (Pay & Display)
You can park free for one hour in our on-street pay and display bays.
The free session can only be used once per day. After the first hour, you can be charged per hour depending on your location.
You must have an active parking session with RingGo.
However, I don't think this position is universal and adjudicators can be fickle: IMO the credibility of your account could still be in play.
The council's evidence shows 2 RingGo entries from ****090. You say you finally registered by Using her [your wife's] phone, I successfully called the registration number and completed the free parking registration by approximately 4:31pm (see attached image) Is their and your evidence in conflict?
You also refer to 'I took a picture of my dashboard with the time and the pcn side-by-side', but these links have expired.
IMO, any formal reps should prioritise the conflict between the contravention description and the council's policy with the 'reasonable time' aspect secondary. IMO, you should also draw them on whether their case is that you had not parked for more than 1 hour but hadn't registered or whether you were parked for more than 1 hour and hadn't paid the appropriate tariff.
Did you have a booking at the restaurant or was your visit ad hoc?