coincidence,
I got asked to clean roller blinds in a conservotary only last week, 16 window blinds and 12 roof blinds,
I tested by misting m/s and towelling to see if any adverse results, plenty of dirt came off and was acceptable for custy, problem was the price wasnt acceptable

Quote " I could buy new blinds for just a little more"

so didnt get the job.
cant understand really I got the suite and carpets to clean

I would imagine that extraction with a cfr or drimaster would do a good job, or a barrier of some kind over an upholstery tool !!!
Geoff