Thanks Ian - I can confirm the problem is not present when using the suggested config, i.e. when using DirectSound.
I dug into the matter some more and found that Audacity forces the Razer Blackshark microphone to Mono when recording using WASAPI. I dug a little into BASS' WASAPI library and can see that the BASS_WASAPI_GetInfo() call yields a BASS_WASAPI_INFO struct which - for the Razer Blackshark - denotes one channel (as opposed to say my logitech BRIO camerea which yields two).
I guess this means that the driver the Razer Blackshark headset is less flexible? Not in terms of having fewer channels, but more in terms of returning less useful data when asking it to record outside its means. I wonder if is there anything to do apart from asking Razer them to make their drivers friendlier?
The simplest course of action - currently - seems to be to fallback to using DirectSound.
In terms of detecting this issue, relying on the WASAPI BASS library appears to be a must, unless you've got any alternative ideas?