I go to Bischofberger Violins to get work done on my violin. I know they have a instrument rental program. They’re located on Capitol Hill on John near Group Health.
I did a quick google search and there’s a Lasley and Russ violins in Ballard that rents cellos as well. They’re new, I think, since I don’t remember there being a violin shop in Ballard by that name when I lived in Magnolia a few years back.