I have now been able to access my records.
An old collectors group (1970s and earlier) with access to pattern books and buckets of buttons in desolate factories named it:
Bristol City & Marine Ambulance Corps, 1890 onwards.
I have learnt that these old boys in the pre-internet days were superb collectors and researchers. I cannot offer any evidence myself.
I agree it has always been an odd button and it would be nice to get hard evidence.
from the web:
Here is the maltese cross on an ambulance:
https://www.ebay.co.uk/p/Oxford-Diec...-43/1011738281