Is it permissible to engage in installment-based buying and selling in a country that imposes usurious surcharges on overdue fines, and do Muslim shop owners incur sin from this, and what are the limits of piety in this matter?

If fines are legitimate and it is permissible to punish financially, then doubling them for late payment is usury (riba). This applies to installment sales where a late payment penalty is stipulated; it is clear usury, whether imposed by the state or the seller. If the fine is imposed by the state and is general in all installment sales, there is no sin upon Muslims, whether merchants or buyers, if they need it, and they must hasten to pay before the fine becomes due. However, if it is possible to avoid this transaction, then it is not permissible to engage in it, because what is forbidden remains forbidden in all circumstances.