Well, Tech Support finally got to the bottom of this for me. I'd sent them the custom polygon data and the report and they had worked on their systems. Then they got me to turn off interactive data, which unfortunately still didn't work. Finally, they enlisted the help of the wonderful Karen Linton and we had a Teams meeting to investigate further. Eventually we found that the GeographicalMappingService stored process was assigned to the wrong application server (our ordinary computer server rather than the VA one). Once we changed that the mapping started working as it should. Apparently that SP is only used for region maps rather than bubbles and coordinates, which is why it was only region maps which didn't work.
So huge thanks to @FalkoSchulz and to Karen: also to Karen Vincent in UK Tech. Support who handled the problem.