Work up new argument for bishop of Rome (7)
I believe the answer is:
pontiff
'bishop of rome' is the definition.
(both can mean the pope)
'work up new argument' is the wordplay.
'work' becomes 'op' (operation, work done in an operating theatre).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'argument' becomes 'tiff' (I've seen this before).
'op' backwards is 'po'.
'po'+'n'+'tiff'='PONTIFF'
'for' acts as a link.
(Other definitions for pontiff that I've seen before include "Head of the Roman Catholic church" , "Religious leader" , "High-priest" , "Bishop" , "Title for a pope" .)