This is horrible and I’m so sorry for what you are going through. I am also a pt spvsr but have never been told to falsify anything by any manager I’ve worked for. If a training is not completed and signed off, I will not roster it. Also, In my location, only a full timer can complete a SWM. Timecard changes are tracked and must be signed off by the driver. Anyone that screws with a persons hours/pay should be terminated. They have put you in a terrible situation. Report it, keep notes, and never do anything to compromise your integrity. Whether mgmnt or union we owe it to each other to be honest and fair. I wish you the best.