The 3 speed SA shift much more cleanly than anything with more gears. I’m only familiar with the Brompton-branded versions - no idea what the SA versions are like.
The current shiney chrome SA lever design is essentially a downtube shifter on an adapter for either bar end or handlebar mounting. So in theory it’s flexible, but I don’t think they sell the adapters separately.
The Brompton branded gears are made by SA and use the same parts throughout as SA hubs with two major exceptions
a) in the BWR gear there is a different planet cage assy and
b) in the BWR gear there is a different driver assy, (but it only differs externally)
The BSR hub is basically a NIG-AW, lock stock and barrel.
Needless to say the Brompton hubs shift the same and behave the same as other NIG-AW type hubs, for the very simple reason that they are essentially exactly the same thing
FWIW the 'chrome' shifters SL-S30 etc you mention are anodised aluminium. They look nice but they stick (in the wrong place potentially) on downshifts (into second gear especially) and (no matter what you do with them, although if you rebuild the shifter with less preload and better lube you can slow down the wear) you get extra backlash as time goes on. The backlash is double-bad news if you try and use one of these shifters with an older (Non -NIG type) AW hub; this type of shifter pulls more cable 2-1 than is necessary for the older hub when it is new, and once it develops backlash, it pulls even more, briefly on every shift. This means the shifter won't latch into gear 1 without pulling super hard on the cable, which is very likely to cause damage and/or put the cable out of adjustment.
These shifters are versatile though; they do fit onto various different brackets, including those also sold with 'SunRace' branding for derailleurs.
BTW there is such a thing an an S3X hub, three speed fixed; this is no longer made but you may find NOS still. This has third gear as direct drive, so is best configured as 3rd = 'normal' with two lower gears for climbing. In truth it isn't a brilliant fixed gear; there is too much backlash to keep purists happy, and, being based on the (2009-2016) 5s (W) models it has some baggage/problems carried over from this design too. You get the same gear ratios as S3X in freewheel form using a 5s(W) hub plus two extra high gears as well. With the 5s(W) hubs the three middle gears are spaced about the same as in an AW hub, so (unlike older 5s SA hubs) there is no benefit interms of having the middle three gears a bit closer together. However second and fourth gear in a 5s(W) hub use an inherently less efficient gear train than first and third in an AW, so there is a small difference there too.
If you are getting a shimano 8s hub then be sure that you get one with roller bearings on the planet pinions; the other sort are pretty horrid to ride on (I think).
If you are thinking of getting a SA 8s hub then I'd suggest that the best home for this would be a bike with small wheels that you never want to ride fast; direct drive is first gear and the gears you use most often are incredibly inefficient, because they all use at least two gear trains in series with one another.
cheers