Maybe you could site the switch inside, perhaps using a pulley wheel
above the soffit.

Thanks for that. It fits with where I'd got to so far which is to (try
to) fit the switch from the loft, in the space above the soffit, with
the cord running through a small plastic tube to serve as a grommet.

Or build the switch inside a suitable external enclosure. Plenty of
those!

I'd wondered about that but was unsure about the entry point for the
cord. If the cord runs freely through the grommet then the enclosure
probably isn't watertight; if the cord is snug in the grommet the switch
might not spring back.........which I guess may be why exterior rated
pull switches aren't common.

I'm grateful for all the comments and sorry to have wasted everyone's
time - although I think there may be a future in Owain's idea of an
illuminated walking stick, especially for when the generation which grew
up with lightsabers starts to need them in numbers.
--
R