How does someone get reimbursed on a do-it-yourself move with no weight ticket?

That is 100% correct. The amount you are paid for your DITY move is based on the weight moved. No weight, no money. They explain it in depth in the briefing and it's usually written in bold in the instructions you receive from the movement counselor.

People learn by pain or repetition. This one is a painful lesson for your Soldier on following directions, but at least it's not an expensive lesson because he will still get paid.

There is nothing that can be done. There is an expectation that you understand what is expected of you when you are in the military. You just don't go figure it out when you get there. This is going to be a tough lesson to learn.

Ask to speak to someone higher up in Transportation. You are setting the Soldier up for a fraud charge if he submits reconstructed weigh tickets. When I moved into the house next door due to renovations, I was allowed to complete and inventory sheet that is used to estimate weight for moves and paid based on that estimate. After I completed the inventory sheet, someone from transportation came over to the house and reviewed it against the stuff in the house. Granted, I applied for the exception before the move but it allowed me to move using the patio doors at each house rather than the front doors which would have been necessary to load and unload a truck just to get the weight. It shortened the distance I had to carry stuff and eliminated the stairs. It might be possible to get transportation to provide a waiver for the weight tickets and make a estimate based on an inventory but you will probably need to talk to someone higher than the person processing the paperwork at transportation.