oH, BOY! You have a problem and part of the problem is that the knotweed has not been declared as an invasive by the state, but maybe your county has, but don't think the county can unless the state does. Guess it is a long process to do that. Maybe a conversation with your neighbor or do you have a "County Agent" or some other person who maybe could drop in on them having noticed their lovely flowers and educate them about the weed. I must admit they are pretty at times, but a total pest!

Japanese knotweed is identified as an invasive species on the DNR web site: http://www.dnr.state.mn.us/invasives/terrestrialplants/herbaceous/japaneseknotweed.html

My first thought would be to look at native shrubs, but they might not grow quickly enough to satisfy the neighbor. You might suggest a combination of shrubs and native wildflowers or grasses - the wildflowers/grasses would serve as a faster growing privacy screen for a few years while the shrubs grow to their mature size and provide a more permanent screen. Without knowing more about the site - sun/shade, soil character, etc., it's hard to offer specific options, but you and your neighbor might explore the options together if they are amenable to your request.

I couldn't help but notice how similar it is to Goatsbeard.

Perhaps that's an option? Not sure if the site works for it or not, but I can tel you that mine springs up pretty quickly in the spring.