Could the late blight pathogen survive between seasons on perennial weeds that it is able to infect (e.g. bittersweet nightshade and hairy nightshade)?
No. This is an obligate pathogen that needs living plant tissue to survive. It only infects foliar tissue of weeds. Since the pathogen cannot infect roots, it cannot survive on weeds in areas where foliage is killed by cold temperatures. Potato tuber is the only plant tissue it is able to survive in.