Please stop communicating with them. You appear to be under the misapprehension that you are dealing with a company that has any inkling of a customer service ethos. Forget that. You are dealing with ex-clamper thugs out to scam you for as much as they can.
Stop sending anything by recorded delivery (or whatever it's called these days). The best that that expensive option is to prove non-deliver if anyone were not to bother signing for it If you are going to communicate by post, all you have to do is go to any post office and request a "proof of posting" certificate. It's free and is enough to use in court, should it ever get that far, that the item was posted and was deemed to have been delivered within whatever the first and second class posting times are these days.
The best and easiest way yo communicate is by email. With email you have a forensic trail to show, to the second, when it was sent. It is instantaneous and if you still insist on writing a letter, why not do it, scan it and attach it to an email as a PDF?
Back to your PCN. Please show us the original NtK you received, front and back showing all dates and times. just redact your personal info, the VRM and the PCN number. Unless otherwise advised, DO NOT under any circumstances reveal the identity of the driver. UKPC have no idea who that is unless the keeper reveal it. Hopefully you did not do so in your earlier correspondence.
The person liable is the driver, who they have no idea who it is. Unless their NtK is fully compliant with PoFA 2012, they cannot transfer that liability to the keeper. More often than not, there are flaws in the NtK that can be shown to invalidate the ability to rely on PoFA.
Everything else in your post is mostly irrelevant. You have not told us the location and why you were parked there at that time if all the stores were closed or closing. Were you parked in a disabled bay? Was the car unoccupied at the time? You say the blue badge was displayed, only not on the dashboard but on the front seat. Please show us the NtK, especially the photos they are using to show that no badge was on display.
The burden of proof is on the operator to prove their case. If the allegation is that no blu badge was displayed, they must prove it. unless they have photos of all the windows of the vehicle, they have not proven that the BB was not displayed. Don't do their work for them.
Luckily, this is only UKPC, a bunch of intellectually malnourished ex-clamper thugs out to scam you. They are not going to accept anything now that they have passed the alleged debt to a debt collector who have added a fake £70 to the original charge. You can safely ignore all debt collector correspondence. It isn't worth the paper it is written and those letters make good kindling to lining for a letter tray.
I will go back over the dates you have given us but be aware, the actual date you receive something is irrelevant. It is the date it is deemed given or served. So, only the dates on the letter or envelope are what count. Fist class post is deemed given or delivered two working days after the date it was sent or issued. Unless you can prove something arrived later than the deed or given date, it doesn't matter.
At this stage, there is little ou can do and writing to them is a wasted effort. You will have to weather the useless debt collector letters. They can be safely ignored. Why you are eventually waiting for is a Letter of Claim and eventually a county court claim. As it is UKPC they will either issue that themselves or use DCB Legal, another verminous group of sub-intellectuals. They good thing about that is that you can guarantee, if you follow the advised process precisely, they will eventual discontinue before it ever gets to court. At this stage of the seedy process, they are hoping you are low-hanging fruit on the gullible tree and will capitulate and pay into their scam at the first sign of threatening letters with scary words like "bailiff" and "CCJ".
So, please answer the questions above and show us the NtK.
