Can I charge a late fee on overdue payments?

Yes, and it is applied by an admin deliberately — never automatically. The add-a-late-fee button only EXISTS on an invoice that is genuinely past due, prefilled with the amount from the school's settings and still editable case by case.

A late fee raises that invoice's total, so it is reflected in the balance quoted by the next reminder and in the billed figures on reports. No late fee is ever born from a background check.