PALS is only for NHS Trust hospitals. However, you could have asked the private hospital to intercede on your behalf if they have contracted the parking operator.
Anyway, it's all a bit academic now. Your company should have simply transferred liability to you by naming you as the driver. That would have absolved them from any liability and a new PCN would have been issued to you, in your name with all the appeals options and deadlines.
Now that the PCN has been partly paid, liability remains with the company. You are being pursued by the company. Any court claim for the alleged outstanding debt will be aimed at the company, not you. The problem for you is that if the company simply decide to pay the outstanding amount, including the fake add on £70, they are probably going to dock it from you.
Any defence of a clam is probably fatally flawed because liability has been admitted (albeit by you, not the company who would be the defendant because they screwed up the transfer of liability). The only argument would be whether the additional amount is a real debt or not.
All I can advise is that in future, should you receive another PCN, explain to whoever in your company screwed this up, that they should simply follow the instructions on the NtK to transfer liability to you and then wait until you receive a PCN in your name and come back here for advice on how to handle it.