How to work with names that don't fit first, last
Many thanks to Andrei for supplying this answer in a comment:

If you navigate to Administer-&gt;Customize Data and Screens-&gt;Display Preferences at the bottom you have two fields:
Individual Display name format: {contact.first_name}{ }{contact.last_name} and
Individual sort name format: {contact.first_name}{ }{contact.last_name}

By updating this setting I can just put whichever is the first name in the first name field, the rest of the names in the last name field, and the records will always show name1 name2 name3 ... everywhere it is listed/sorted.
The last step was to update the existing entered names. I did this with the following SQL, which was appropriate for my particular use (and may not be for yours!):
UPDATE civicrm_contact SET sort_name = display_name
WHERE contact_type = 'Individual';

And now my names are working exactly as the client requested throughout the system. Many thanks.