I'd say it is gehoben und literarisch, but not gewollt altertümlich. I might even use the construction myself on occasion, in written English. Note that the text also employs the subjunctive, the register is consistent.

I would say gehoben, literarisch, and at least pretty stuffy / pretentious, if not quite gewollt altertümlich.

Someone writing modern English would probably say something more like 'unclear but (in any case) irrelevant.'

But the BE writer gets full marks for the subjunctive!

'... asked that Senator McCain be supplied with ...'

*applaus*

From my legal studies (many years ago now), I vaguely remember reading "matters not" in a number of English law reports. I can imagine Lord Denning readily using it, for example. Perhaps that has influenced this High Court judge?