Thanks @Trenloe and @Marqeis_de_Taigeis in particular. I tried the Virtual Cable software as it seemed to be the simplest and setting it up was. I tried with the browser version of Discord initially and called myself from a secondary account I created for this. It worked, but the sound quality was poor. As well as being a bit patchy in general it would suddenly change volume and at times cut out completely. With the browser version of Discord it would only follow whatever the default sound options I'd set for Windows as a whole and I wondered if the browser version might be inferior so I installed the Desktop client of Discord and tried again with that. This version also lets you pick specifically the virtual "Cable" input and output devices which I did. However, the sound issues weren't solved. Still the unpredictable volume changes, cut outs and overall fuzzy quality.

I haven't tried either of the other two pieces of software yet but will if I have time. However, I have a sinking feeling that the problem is on the Discord side of things rather than the specific software. FWIW, my Internet connection is high bandwidth and ought to be plenty sufficient.