[YPPH]

"Not common sense" is not the legal test for patentability. But it is in effect incorporated within the requirements that a patentable invention be novel, have an inventive step, and be non-obvious (assessed without the benefit of hindsight).

The US Government could dispute the patent claim if they think the so-called invention is not an invention.

[rkagerer]

What stage in the process of disputing the patent's validity are they at?

Are there higher judges they can appeal to for a more rational outcome?

[YPPH]

Patent disputes in the United States arise in federal jurisdiction.

In relation to a patent infringement matters commenced in the US District Court of the relevant judicial circuit, it can be appealed to the Court of Appeals. In theory it can then be appealed to the Supreme Court of the United States (as happened in Google v. Oracle), if that court thinks its intervention is warranted.

In relation to issues arising from the grant of a patent (not an infringement action), an appeal can be made to the Patent Trial and Appeal Board, and then to the Court of Appeals for the federal circuit, and then to the Supreme Court of the United States, if that court thinks its intervention is warranted.

I read below this is working its way through the appeals process but I'm not sure about its status.