I have finally found the answer. The clue came from pcguru's answer beginning 'Since the server has only 1 GB of RAM...'.
On a hunch I looked to see whether Apache had memory limits of its own as those were likely to affect PHP's ability to allocate memory. Right at the top of httpd.conf I found this statement: RLimitMEM 204535125
This is put there by whm/cpanel. According to the following webpage whm/cpanel incorrectly calculates its value on a virtual server... http://forums.jaguarpc.com/vps-dedicated/17341-apache-memory-limit-rlimitmem.html
The script that runs out of memory gets most of the way through, so I increased RLimitMEM to 268435456 (256 MB) and reran the script. It completed its array merge and produced the csv file for download.
ETA: After further reading about RLimitMEM and RLimitCPU I decided to remove them from httpd.conf. This allows ini_set('memory_limit','###M') to work, and I now give that particular script the extra memory it needs. I also doubled the RAM on that server.
Thank you to everyone for your help in detecting this rather thorny issue, and especially to pcguru who came up with the vital clue that got me to the solution.