Answer
Check the individual field-level LOV checkbox on the Custom Logical Unit's own fields, not just the persistent custom field that references it — each field on the CLU that you want visible in the LOV popup needs its own LOV checkbox enabled. Once that's set on the relevant fields, you can then configure how those fields appear (order, visibility) directly within the custom field's own LOV configuration settings, and all four fields become available in the search popup rather than just the one that was previously flagged.