Hi Julie
We did this 20 years ago, so our info may be out of date. We hired an agency to do it for us, and it was expensive, but they took care of all the paperwork, had the dog signed off as fit to fly at one end, and fit to arrive at this end. We also had to pay some duty and we actually collected her from the customs sheds here.
Maybe check with TUI and ask what other documentation is needed.