Is it permissible to pay the driver's fare—for a driver who left with the questioner's bag and did not collect his fare due to forgetfulness or to avoid a traffic violation—to another party?

If you are unable to find the owner of the taxi, there is no sin upon you for not paying the fare, and you should waive your right. If you despair of finding him, you must return the money to its owner or his heir. If there is no heir or his news is قطع (his whereabouts are unknown), then pay it to the Imam to place it in the public treasury (Bayt al-Mal), or to the ruler authorized to manage public funds. If that is not possible, then donate it with the intention of [repaying] the debt, or spend it on public welfare projects. If there is an unauthorized judge, you may give it to him to spend on public welfare, or you may distribute it yourself for public welfare, or you may hold it until Bayt al-Mal is established. In summary: if you are in a country where the rulers spend funds on public welfare projects, then hand the fare over to them; otherwise, spend it yourself on a public welfare project.