Do not so anything else until you fully understand how this all works.
First thing, do not communicate with the debt collector. They can safely be ignored. They have no power to do anything, no matter how scary they try to be. Honestly, do not contact them again. Ignore them.
I would place money on the fact that Civil Enforcement DO NOT own the car park. They are simply contracted to operate it on behalf of the landowner or the landowners agent. You need to try and establish who owns the car park. Possibly the hotel. With who did you have to register the VRM for an ePermit when you moved there?
When did you change the address on your V5C? Changing your address on your drivers licence does not automatically update your V5C. If it was at around the time you moved to the hotel, CE may have your old address. The debt collector probably did a soft trace.
You need to confirm this with CE by sending a data rectification notice to their DPO (email address in their privacy notice on their website). You must
instruct them to
update their records with your
current address for service and to
erase the old address and to
confirm with you when they have done so. You do not want them to hold two possible addresses for you. That is exactly how many people get default CCJs. You need to be able to receive all correspondence on time.
You mentionphe "two letters". These are for a single PCN. They are the original Notice to Keeper (NtK) and the reminder letter. There aren't two separate PCNs, especially as you only mention £170. The PCN was for £100 and the debt collector has added a fake £70 fee.
As CE are BPA members, you could complain (not an appeal) to them and request that they re-issue the PCN because you never received the originals. There is a clause in their Code of Practice (CoP), 23.8 that you can try and use that states:
You must have a process for considering appeals received outside of the normal 28-day period allowed for lodging an appeal where the appellant provides evidence of exceptional circumstances for the appeal not being lodged within the normal timeframes – where the addressee only discovers and can show that a parking charge notice has been issued in their name after the 28-day period the period must restart and any enforcement, excluding court action, must be paused.
If they don't you can then complain to the BPA that they have not followed the CoP.
For now, find out who at the hotel is responsible for adding your vehicles VRM to the "whitelist" and why wasn't on that list when the PCN was issued. Ask them who contracted CE and if it was the hotel or an agent of the landowner, you must ask them to get the PCN cancelled.