OP, I'm not certain how registering after the event - and thereby obtaining 3 hours MORE - helps when the maximum permitted period is 3 hours.
Don't underestimate the authority's resolve to deny that the contravention grounds are wrong by latching on to, as they might see it, a deliberate attempt to park for more than 3 hours i.e. park, don't register, register after any PCN and then claim that registration was made.
IMO, stick to:
1. Contravention did not occur;
2. During an observation time of a maximum of 2 minutes, but probably less, the CEO could form a considered belief that a contravention had occurred.
1. That there is no charge payable to park in the car park for up to 3 hours is unarguable: it's stated clearly on the tariff board i.e. '3 hours free'.
The contravention grounds must therefore be incorrect.
If, as appears to be the case, the CEO thought that registration for 3 hours' free parking had not been made then it follows that the correct description would be 'Parked for longer than permitted' because absent registration the driver is not permitted to park.
The PCN must therefore be cancelled.
2. On the subject of the CEO's belief after no more than 2 minutes, I would refer you to the photos showing service of the PCN and the tariff board. The former is timed at 11.47.30, the latter 11.48.38, an elapsed period of 1 minute 8 seconds. It will not escape the authority's attention that if the CEO took more than 1 minute to walk to the tariff board simply to photograph it then it would take a driver at least twice this time and more likely 5-6 minutes to walk to the board, read its contents thoroughly, see if they had the wherewithal, in this case a smart phone, download an app(often problematic), read and understand its conditions, particular the need to enter correct VRM and location number and then comply. To this must be added the time the system takes to be updated by the app.
2 minutes is wholly inadequate and therefore the PCN must be cancelled on the grounds that the 'penalty exceeded the amount applicable in the circumstances of tbe case'.
OP, I've deliberately made this two points because they won't want to acknowledge the former and therefore could save face by acknowledging the second.
Are you the registered keeper with current DVLA details?