Free Traffic Legal Advice

No offer of discount, so it is now a total no-brainer to take them to London Tribunals.

Please post a GSV link to the exact location where you parked. Photos show you seem to be straddling two bays, so we need to see the bay.

Well, first off, you are not "formally appealing", you are submitting an "informal challenge". It's important you know the difference because if your challenge is declined, like over 95% of them are, the next thing you will receive is a Notice to Owner, (assuming you are the owner), against which you can submit formal representations.
Representations against an NtO are usually looked at a lot closer, because councils just want the easy money, so you might get it cancelled at that stage, but many councils know that they can "DNC" (Do Not Contest)  an appeal at the TPT having put you through a lot of hassle. OK they lose the case, but the enormous sums of money councils earn with PCNs means they can lose a few.

Secondly, it is almost inevitable that the location isin a Special Enforcement Area, because any council that has powers to enforce parking and traffic contraventions, must have first established a Special Enforcement Area in order to do so. This takes responsibility for enforcement away from the police and hands it to the councils under the Traffic Management Act 2004, plus associated regulations. Virtually the whole of England and Wales is now under decriminalised parking and traffic enforcement.

Of the points in your draft challenge, only 3 and 5 have any value. The disused entrance does not meet the statutory requirement for enforcement, because it serves no purpose as entrance or as a road crossing, so they really should cancel this PCN solely on that basis, but they won't because they know that on receipt of a rejection, most people, as I say above, just cough-up. If you want to succeed, you have to forego the discount option, wait for the Notice to Owner, send in reps against that, get rejected again, then register an appeal at the Traffic Penalty Tribunal. Are you prepared to do this ?  If not, it's game over now.

I have a long saga with a PCN after driving in a Bus Lane. Please bear with me as I upload details. This is new to me!

How long must we "bear with you" ? Better not to start a thread until you've got all your ducks in a row, surely ?

You've blanked out too much on the PCN. You need only blank out Name and Address. We need PCN Number and car registration to see the video.
You might also care to read this: -
https://www.ftla.uk/civil-penalty-charge-notices-(councils-tfl-and-so-on)/read-this-first-before-posting-your-case!-this-section-is-for-council-tfl-dartme/

AFAIK, we've seen this location before, looking at your GSV link

So, postal PCN received but ignored, then CC arrived adding 50% onto the PCN penalty.

Frankly, there is not a great deal you can do about this. If you wait until the PCN is registered at the Traffic Enforcement Centre, adding another £10 onto the amount, you can submit a Statutory Declaration, but none of the three circumstances you can select from on the form match yours. Your only option is pay-up. Sorry if this isn't what you wanted to hear, but if a recipient of a postal PCN ignores it, then any appeal options are lost.
See here, Form PE3
https://www.gov.uk/government/publications/form-pe3-challenge-an-unpaid-penalty-charge-notice

Edit
I suggest that you contact the council to explain the situation; they may cancel the CC as a courtesy. Your mother can then pay it, or submit representations.

Until we know what is there, it is difficult to give advice.

I had a look at this junction on GSV and there are no directional arrows on the last view of July 2024, so this looks like a fairly new restriction. The only thing there is the two traffic lanes on Peckham Rye here: -
https://maps.app.goo.gl/VZts8teH9NtuxiMn7
They may have put in new signal heads that split the green phase for the two lanes.