If I had more than one run of SWA I would be tempted to put up cable tray and clip to that. If the tray is vertical (parallel to wall, supported off by short lengths of strut, for example) then you need the occasional metal tie for fire support in addition to regular ties of any other sort.
If the SWA sits on top of a horizontal tray (so it can't fall if clips melt), for example supported on cantilever arms, or suspended from a roof, then standard cable ties are OK. Personally I would always use the black UV-tolerant type even indoors as sometimes it will get sunlight from windows, etc, and it saves me keeping multiple bags.
However, I also get the feeling this person is being difficult for no obvious reason. I would be tempted to present your suggestions for alternatives in writing (e.g. email) and make sure they respond also in writing just to cover yourself.
Incidentally, when you said trunking I was thinking of the metal stuff, 50x50mm upwards, not plastic. Can you confirm the type?