Yes, apparently the plumbing of the rose runs more vertical than horizontal, with roots on one side more connected to the canes on that side. You may have to dig the plant and bare-root the part you are trying to save. It is worth trying this if the crown is splittable, but, with those own-roots that make a tap root, you can't split it. Perhaps you could get a bit of root with a cane stub and treat it like a cutting.

I tried splitting twice and it didn't work, but I would try it again. I drove a hatchet blade with a heavy hammer to divide the plant. The plants survived but RRD symptoms recurred.

I'm trying this with the mini 'Kismet'. Basically disassembled it and replanted the section that appears nonaffected by RRD. We'll see what happens next Spring.