The bottom line is any appeal based on the defence given will be rejected, Parkingeye won't accept their PCN isn't POFA compliant. An appeal to POPLA is also unlikely to be successful. This is not because either body routinely reject appeals, it is because the defence relies on an interpretation of the legislation that even the experts on here disagree on.
After that you will get lots of letters from debt collectors. Eventually you will get a letter of claim. At that point it may go to court. I don't think we know of a case won on this defence, although that's often because cases don't reach court or there are stronger defence arguments.
The driver didn't follow the car park rules and unless you as the keeper name the driver they can come after you for the PCN. Not paying the invoice is a breach of contract they can successfully pursue you for.
Parkingeye are known to be more litigious tha many private parking companies. The real question is are you feeling lucky and are you prepared to engage with the court process?
I didn't realise the prospects at POPLA were so bleak. I found some decisions that I thought were helpful, but to be clear you're saying they are not?
POPLA Appeal 6060637496 (2016)
POPLA Appeal 6061796103 (2016)
POPLA Appeal 6160338204 (2017)