I'll update this thread, in case it helps someone else. The ticket I opened allowed cPanel's support to track down the problem. It was quite simple in the end, but I was overlooking a couple of details.
In my case, it was to do with the amount of memory that the cPanel processes are allowed to use. Setting it to a higher value solved the problem. There were two reasons why changing this setting did not appear, at first sight, to solve the problem.
1. It's about the amount of memory that cPanel is allowed when Easy Apache is run, not the amount of memory available when viewing the "Configure PHP and suEXEC" screen. Having increased the amount of memory, the next step is to re-run Easy Apache and then come back to that page.
2. The easiest way to change the amount of memory allowed to cPanel is to change the setting in the Tweak Settings interface in WHM. Usually, when a configuration is changed in Tweak Settings, WHM notices what settings have changed and automatically makes any other alterations needed to make the change "stick". In this case, the cpanel daemon needs to be restarted, but WHM does not do this for you. (Arguably, this is a minor UI bug).
So the correct order is:
1. Go to Tweak Settings. Increase the amount of memory cPanel processes are allowed.
2. At the command line, run /etc/init.d/cpanel restart.
3. Re-run Easy Apache.
4. Go to the Configure PHP and suEXEC screen. The drop-down boxes should be populated.