Not sure if this is what you are after, but there is an option 'displaynameFormat' you can use in the refinery parameters. https://manual.collectiveaccess.org/providence/user/import/mappings/refineries.html?highlight=displaynameformat#entitysplitter
There is also a setting 'defaultDisplaynameFormat' that you can set, which will format the displayname for all entities created. The setting seems to have the same values as the options.
If you want to format it based on something completely different, try in the refinery parameters to set 'preferred_labels.displayname' with reference to the source data you want, using the ^ notation.
I can't recall 100%, but I think when doing an entity splitter, it takes the source data and makes that the displayname, trying to extract when it thinks is the first and last name from that if not explicitly set. So it may also be possible to use in the options 'formatWithTemplate' to create the display name before the refinery processes it.
https://manual.collectiveaccess.org/providence/user/import/import_ref_formatwithTemplate.html