The U.S. court recognizing the foreign judgment need not have jurisdiction over the contract dispute itself. It need only recognize that the foreign court had jurisdiction over the case and the parties.
Actually, there are two types of jurisdiction--personal jurisdiction and subject matter jurisdiction. Because the terms of most employment contracts expressly consent to the jurisdiction of the foreign court over the parties, personal jurisdiction exists and is not an issue, even if the pilot is not a resident of the foreign court's forum.
Subject matter jurisdiction is the power of a court to hear the dispute. For instance, in the U.S., federal courts are courts of limited jurisdiction. They can generally only hear cases that deal with federal questions or where the parties are "diverse," specifically, where the defendant is not a resident of the state and the amount in controversy exceeds $75,000 (known as "diversity jurisdiction"). On the other hand, state courts are courts of general jurisdiction. Because of this, most contract actions are heard in state courts.
It is unlikely that a foreign carrier's lawyers would be foolish enough to file their dispute in a local foreign court that lacks power to hear the dispute. There is certainly a correct local court in that foreign jurisdiction to file the action and I am confident they will file it there. In addition, there is a forum selection clause in the contract that specifies the proper forum and a choice of law clause that specifies the law to be used.
Once they get a default judgment from that court, then it becomes a question of taking the judgment to a U.S. court and domesticating it.
The lack of jurisdiction issue is not a question of whether the U.S. court has jurisdiction, it is whether the U.S. court, when presented with the foreign judgment, would determine that the foreign court entering the judgment had jurisdiction. Of course, for the state court to domesticate the judgment, it would have to have personal jurisdiction over the pilot (e.g., pilot resides in that state).
Since the employment contracts I have seen generally contain forum selection clauses and consent to personal jurisdiction of the foreign carriers' courts and laws, I am looking for information about cases where foreign carriers have actually tried to enforce a foreign judgment here in the U.S. Unlike federal courts, most state courts' records are not available on-line, unless the case actually got appealed and a decision from the state appellate court was published. Thus, there are probably few (if any) cases available on-line.