Binks - pretty rare in this period
AMAC - pretty common in this period
Villiers - easy to find, but you'll have trouble convincing others that it's correct! In the early 1920s James used the Mills carburettor, which was adopted (bought out?) by Villiers, so by 1928 a "mills" carby hadn't changed much, other than being re-branded Villiers. I think it was still bronze in 1928??
I assume your bike is a 500c side-valve twin, so the required carby is only small (250cc-size). Does it bolt on to a stub or a flange?
Binks would be a model J (2-lever) or JA (single lever) (7/8") or a model K or KA (1").
AMAC would be a model 30M (7/8") or a model 25M (1"). In 1928, all AMACs have the remote air slide on the side of the mixing chamber, so many people will think they are "TT" carbies. Not a problem, except that the price goes up!
Not sure about Villiers, but anything with a brass body and a slide diameter of 7/8" or so would probably work OK.
Cheers
Leon