After further testing -- it does look like putting the laundry line on a covered porch does seem to work, even if you use a regular ceiling rather than a glass awning.
(It took me a while to figure out how to make the ceiling though! I had never made a covered porch without a room over it before.)
Edit to add -- I live on the coast where it's really foggy, and I have a clothes horse too (like the 3rd from the top left in the picture above!) -- I put it on the porch when the weather is nice but inside by the heater when it's not! Seems like our sims definitely need something like this!
