TiggR_R1 - Thank you very much for these photos they are very useful: they have not solved my problem

, but put me in a much better position to solve it.
I had naively transferred the relays from my facelift donor to my prefacelift Z3 expecting them to work

.
I have now inserted relays matching your description/photos, and which now correspond to the BMW schematic diagrams for a late 1998 yo March 1999 Z3.
Electrical circuits are not my strong point, though I can follow it in principle and run basic voltage and continuity tests.
Althulas’s comment regarding the incorrect wiring is something I had read and was able to set aside as my plug is the same as you show (photo again useful).
So, now all the pointers are to either the Aircon controller or wires leading to and from it.
Now that zi know I have the correct relays in place my testing becomes meaningful.
(Previously with the wrong relays it was a case of “Rubbish in: Rubbish out!”)
ChstGPT has been very useful as a logic calculator and advisor on what, and how to check the next process.
The search continues……..
But thank you both once again for your help.