I empathize with you.
I have learned from here the following plan of action:
1. Without revealing identity of driver to the Private PCN issuer, try to get it cancelled via the establishment you were a customer of, this option is long past for you at this stage.
2. If #1 is not working, post the NtK (Notice to Keeper) picture here and wait for advise. In majority of cases, this notice isn't PoFA compliant (Google or search posts here for details) which means if they don't know driver they can't have keeper liabile. So the advise is to refuse sharing driver identity, challenge as a keeper only and force them to cancel PCN.
3. If most cases, they still cancel the appeal made as per #2 (as they don't even read appeals and only care about ££) but they will give you POPLA code. So in that case get a draft from here and without driver identity you very likely succeed at POPLA.
In your case #3 is also already past.
Now if your feeling you are in the right you have no option but to wait as you can't appeal or challenge this at this stage.
After 28 days have past you will start getting intermittent letters from DCBL or similar company for £170 payment. These would sound scary and will try to threaten you to pay. They have no power, don't waste your time on them and don't engage with them as they are only interested in your money and nothing else.
Eventually, they will either discontinue sending these letters (in my case from before I learned about this forum they did) which is win OR you will get a " Letter before action" or some actual legal document. That's when you would need to respond/defend and you can come here for advise then. I have been told here that even if you lose eventual court hearing (which might not even happen) you would only pay £100 and not £170 as DCBL will threaten you with. So I suppose best advise now is to wait and learn a lesson to post NtK here if you get one again.
More knowledgeable people can comment/correct me on above please.
Best wishes.