Fair enough, but 4k runs on HDMI and it does say 1080 in the description.
Makes me wonder if this 4k-HD distinction is akin to 'digital' vs 'analogue' aerials. Though 4k needs higher bandwidth, so maybe a 4k device needs a more capable chipset.
In simple terms* "4K" - which usually means 3840x2160@60Hz - requires HDMI 2.0, especially if HDR is involved. Earlier versions of HDMI did support 4K resolutions, but at lower frame rates. HDMI 2.0 was released in 2013 so significantly post the release of the HDR FOX-T2. I suspect that the HDR FOX-T2's HDMI port is version 1.3, or possibly 1.4.
In simple terms (again): given the HDR FOX-T2's known 'fussiness' about its HDMI connectivity, it would not seem unreasonable to expect/hope that connecting it to a device which uses the same or a similar HDMI version might at least eliminate one way for it to decide not to play ball.
I've now ordered both of the devices available from Amazon that I mentioned previously (the audio extractor and the HDMI splitter - the latter having a 50% discount offer until 7th December, woo hoo) to try. I'll give the splitter a go first and if that does the trick then the audio extractor can go back unopened. If neither of them work then I think I will almost certainly give it up as a bad job.
(I know I said I wasn't going to get sucked in to this but the urge to try to get things working can be difficult to control...)
* Because, like almost any 'standard', especially ones which allow selective implementation, it turns out to be not very simple at all: https://en.wikipedia.org/wiki/HDMI