Foreign National Firearms Background Check Enhancement Act This bill modifies the exceptions to a provision that generally prohibits a nonimmigrant visa holder from obtaining a firearm or ammunition. Current law provides certain exceptions to this prohibition against a nonimmigrant receiving a firearm or ammunition, such as for a nonimmigrant who has a U.S. hunting license. This bill narrows the exceptions by, among other changes, requiring a nonimmigrant with a hunting license to get a waiver from the Department of Justice (DOJ) before receiving a firearm. The person transferring the firearm to a nonimmigrant visa holder must verify the validity of the waiver with DOJ.