Thanks Eric - at this point, I appear to be running out of options in any event.
Here's what I have managed to do over the weekend, after a number of issues arose:
After the advice above, I realised that I that the actual problem was not where I was trying to erase the drive from, but rather, that the drive was refusing to unmount.
Regardless of which drive's Recovery Mode was being used (and this included a bootable USB HS installer, AND the Internet Recovery Mode – the HDD would not erase since it could not be unmounted.
Finally, after surfing around, I was pointed towards using Terminal, where I managed to "force" unmount the HDD, and after re-mounting it, I managed to erase the HDD.
In other words, after booting into the SSD, I simply ran Terminal - forced the unmount, and could then erase as per normal.
It is now working as expected - and is seen as an empty drive on the system, that I can copy data onto.
I want to point out at this point that First Aid in Terminal confirmed that both SSD and HDD are functioning properly – with the exception of whatever was clinging on to it, keeping it from unmounting.
I then tried to upgrade the SSD drive, to High Sierra - and lo and behold, the same issue arose. It then dawned on me that this was the same issue that was preventing my upgrading to HS, when I only had the HDD – something I had tried periodically over the past few months.
The installer, when it was trying to work its magic, could not unmount the drive mid-installation, and so stopped the upgrade process.
There are other threads on the Apple Support Forum, where this is explained in detail - unfortunately, the solutions that worked for other's over there, were not working for me.
Since I can 'force' unmount the SSD, but the installer cannot, I am stuck - and cannot upgrade. Obviously, with me using the Migration Assistant to transfer the data off the HDD over to the SSD when I installed the latter, has seen the underlying 'mount' problem transferred across as well.
So it would appear that if I have any hopes of getting my Mini onto HS, as my MBP is, I will need to wipe it and start over.
I am very cautious about that though, since the Mini is the repository of my canonical 248GB Photos library...
So want to try and keep it intact if at all possible, or at least, be able to restore from the TM backup.
If I do wipe, and upgrade the SSD/Mini to High Sierra, I should be fine to restore from a Sierra backup, correct?