I am at a loss of what options I have.
Procedurally not many.
To date:
Avis received a PCN issued on 31 May which they paid.
Subsequently, Avis wrote to TfL by letter* dated 16 July to authorise you to make representations.
You submitted representations.
TfL responded by letter** dated 27 Aug. that you were not the addressee on the PCN and that they had written to the addressee giving them 14 days to confirm that they had given permission for you to act on their behalf.
Clearly a c**k-up on TfL's part.
*- address used by Avis: Travel for London(??), Red Routes, PO Box 335, Darlington, DL1 9PU
**- address on TfL's letter: Transport for London, Red Routes, PO Box 335, Darlington, DL1 9PU.
When you get back to TfL stress that Avis had already written to them a month before they replied to you saying effectively that they hadn't..and here's a copy of Avis's letter to TfL. NB. Avis didn't write to you, they wrote to TfL and merely copied this to you.