You are the “lessee” or “hirer”. The Leasing company is the “lessor”. You need to get the terminology correct as you are dealing with contract law.
The fact that the lease company has paid the PCN thus depriving you of any opportunity to appeal is unfortunate. However, it will depend on what it says in the lease agreement about parking charges from unregulated private parking companies. These are not “fines” or “penalties” issued by any “authority”.
If the lease company is a BVRLA member, they must have a specific clause in their wording in order to absolve their liability if they have paid the charge in breach of those terms. Can you show us the actual wording in the lease agreement about paying charges from private parking companies?
If they have paid a private company under the excuse that it was a “fine” or a “penalty” without the lease agreement specifically allowing them to pay invoice from private companies, you only recourse is to complain to them, the BVRLA and if they do not refund you, to take them to small claims court, an easy thing to do which will be explained once we’ve seen the specific lease clause.
Under no circumstances shoul the hirer identify the driver. Neither the lease company nor the PPC knows who the driver was. Any communication is as the hirer/lesee. If the lease company had followed the correct procedure of simply transferring liability to the hirer/lessee, this could have been ended and you wouldn’t be paying a penny. I have yet to see any PPC correctly serve an NtH that complies with PoFA. The actual NtK you’ve shown us is not PoFA compliant anyway and the lease company could have easily defended it, but that’s besides the point.
For now, please show us the exact wording of the lease agreement to determine the best course of action for now.