The bsa pre unit singles are a fair bit lighter than equivalent twins, I’ve a b31 rigid and a10 s/a, you can get an unexpected thump in the rear from rigids from time to time, but I find it’s only really a BIG thump above 50. You do need to dodge pot holes at low speed of course.
A viable alternative could be a c11? I’ve got a c11 rigid one I’m “re-commissioning” as a sideline project (saw it at auction selling FAR too cheaply, attending auctions to “just look” is dangerous!

). I’ve noticed the c11 shares quite a few parts with the b series, is very similar design, and is lighter still, and it’s easier to kick start. Another advantage is the coil ignition so no potentially pricy magneto issues

to deal with.