Difficult one to call, this.
We now see lots of PCNs about payment via mobile phones as councils ruthlessly cut out ticket machines. Clearly the council didn't lose any money because you made a payment, albeit much later in the day. Their letter is the usual Fob-Off letter we see all the time, but at the moment, I can't see a good appeal argument at London Tribunals. I say this, because if you decide to wait for the Notice to Owner, it is on the understanding that you are standing your ground and are foregoing the discount option. Reps against a Notice to Owner are looked at more carefully than the people who send out the Fob-Off letters, but even so, most are refused, (like circa 90%), so you end up at LT with the full PCN penalty in play.
There may be a 'technical' appeal argument based on Islington mismanagement of the enforcement process, so wait a bit until others have commented, but don't miss the payment or appeal deadline on the PCN. Recent technical appeals have succeeded at London Tribunals on the issue of the webpage texts making unlawful demands for payment.