It is a basic principle of parking and paying for it, that time is allowed for making the payment.If it wasn't, nobody could park legally ! If the council force use of a payphone app, then they must accept that many people won't have the app on their phone, and must download it, then invoke payment. All of this takes time, so if it were me, I'd submit a challenge on the basis of the above.
The council will likely refuse your reps so you may have to take them to London Tribunals were I think you stand a good chance of winning. Do be aware, though, that if you do this, the full PCN penalty is in play. It is a double-or-quits gamble, essentially.