I'm not aware of it invoking, but a careful read of the 13th Amendment will find there's an exception to the slavery ban:
Neither slavery nor involuntary servitude, except as a punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.
This means that prisoners can be forced to work for the government for no pay if they are duely convicted for their crimes (often seen as a chain gang, which you can see from countless films, mostly exist so someone can escape from them in the story). Obviously these days the issue is largely moot as most prison work programs pay the prisoners, though this may be cents for labor otherwise netting above minimum wage, and chain gangs are no longer used (work details beside the roads are usually paid and its only for model prisoners. It's a treat for good behavior, not a punishment).
So while prisoner slavery is not used in modern society, it is still legal in the U.S. and thus the Fugative Slave Clause is still law and can be used in the modern setting since contrary to modern thinking slavery isn't 100% illegal in the United States... it's just most of the parties who could use this one type of legal slavery do not choose to use it, it makes the Article IV clause largely irrelevent. Should a state decide to exercise it's right to prisoner slavery, then they have the fugative slave act to back up their extradition of an escaped prisoner.