They actively disallow it? That is rather, umm, shocking? If anyone ever called out the "if you've got nothing to cover up..." line, then I'd say it fits here.

Has anyone worn a film badge dosimeter while getting scanned (or just put on in their pocket)? That would be a way to independently test the radiation dose.

If everything is configured properly, the dosimeters would read normal. If not, isn't it better to find out as soon as possible?

No. Imagine if a few workers were being overdosed. The TSA would get sued and they'd have to find some other way to annoy us.

No data, no lawsuit.

Using dosimeters works in many places where they use equipment that produce X-rays. On occasion, an incident happens. This may lead to a lawsuit, but still a lawsuit much smaller in scope than when workers are being overdosed over longer periods of time and them finding out in a few years.