Long shot (which I'm also going to try with the likes of Eastside, even though it was their miscommunication with Mercian that lead to the part being missing in the first place, and Soul Cycles)...
Have had the calipers out and the BB shell on the Sunbeam appears to be 31.5mm (or thereabouts) which is around 1.25". Usual size is between 34 and 35mm, or 1.375". Which means sourcing a non-drive side cup is proving difficult (the drive side is fine as it's basically part of the frame). Not even Sheldon has anything about 31.5mm BB shells.
Still keeping an eye out for donor Sunbeam frames on the likes of eBay (and still kicking myself for forgetting about one I was watching that went for only £32).
But, just in case anyone at all has a weirdly small BB cup sitting about the garage/shed/kitchen drawer, I'll take it off you!
(headset bodge isn't perfect, but works, but I'll still be keeping an eye out for the old 'Headclip' style that came off and was also 'misplaced').
Put my new Soma Lauterwasser bars on last night and was able to scoot around the garden. Feels so close to being finished, until I think about everything still to sort!