Is it permissible to exploit an employee's predicament to recover money he unlawfully took, given that the legal right to claim it has lapsed due to the statute of limitations? And does the right to claim lapse due to the statute of limitations in Islamic law?

The forfeiture of a right due to prescription is not from the law of Allah; rather, it is a prevention of the judge from hearing a lawsuit after a specific period, under certain conditions. Invalid transactions do not become valid with the passage of time, and the ruler's judgment does not permit what is forbidden nor forbid what is permissible. Thus, a right does not fall due to the passage of time, and the non-hearing of a lawsuit after a period has passed does not mean the invalidity of the right. If the questioner has a proven right against the employee, and the law did not assist him in recovering it, then there is no blame upon him in recovering it without transgression if he obtains it and if no greater harm results from it than the harm of the right being lost.