5 minutes grace period didn't seem enough.
It's a consideration period and 5 minutes is standard so not really an avenue for an appeal. A grace period applies if the driver paid to park and then overstayed.
A quick pootle around the car park on Google Street View shows 2 payment machines in the car park, did the driver drive in, park, check both payment machines, attempt to use the app and then drive off all within 7 minutes?
I'm not suggesting they didn't just checking you are 100% sure there were no ways of paying on the day. They will know if other people used the payment machines or paid by app on the day which they will be able to use as evidence payment options were available.
So as grounds for a successful appeal, no I don't think you're likely to convince Parkingeye to cancel the PCN or for POPLA to uphold your appeal either on these grounds.
The usual advice is that if you really don't want to pay the invoice you can probably run the clock down by going through the process as most cases get dropped just before they have to pay the court fee. However the process will take months, you will get hounded with debt collectors letters and you will have to file a court defence at the right time. Worst case is it goes to court, you lose and it'll cost you 4 to 5 times the original invoice, or if you fail to pay then you'd get a CCJ which would seriously screw up your credit score.
Up to you whether you want it to go away quickly or you're prepared for the long haul with them probably but not guaranteed to give up at some point.