Interlining the bags should not be the problem - 3 bags may be. Your bag tag numbers will not carryover to Jet. When you go to checkin with them, they will ask if you have bags to check. At this point, you will have to give them your tags so they can input the numbers in their computer. They will see you have 3.
Others in this forum are probably more experienced with Jet and how they will handle things from here.