I am not familiar with the exact equipment fitted to your aircraft. We had a different setup on our 320 series.
From your discription I would say it would be correct to use the (your words) "upper ATC button" in AUTO during ground ops from push back to take-off and after landing.
I guess it would then be logical to use ON when lining up and in flight.
The switch you describe on the TCAS panel with STNDBY-TA-TARA positions seems to be for the operation of the TCAS system only, I don't think it controls the transmission of transponder SSR / mode S signals.
What does your FCOM say about the ATC panel and function of the switches? If youre SOP's are not clear on how to use the equipment, ask your tech pilot's/fleet manager to clarify and maybe adjust the SOP.
It seems that more of your collegues are unsure how to operate the transponder in the case you describe.
Grts TS737