Well, we have finally come to at least a temporary conclusion to this problem. I took the bamboo bull by the horns one day and talked to the neighbors. At first it was a total fail and they would have nothing to do with taking care of the problem. "Not my problem."
They must have had second thoughts, however, as one day out of the blue they asked to bring a "specialist" over to see our yard. As it so happened, there were a bunch of shoots I had not yet cut down that week and a giant one making its way horizontialy across the ground.
Success! They admitted it was a bigger problem than they realized and said they would do something about it (even if it was not exactly the solution I was hoping for). This "specialist" came in and took out the shoots on our side, some of the bamboo on their side, and dug a trench about 8 inches wide on their side of the fence. They said they would see that no shoots went through the trench.
Well, we will see what happens. I don't think I'll really know until next Spring. I have found a few small shoots on my side, but since they at least made an attempt at fixing the problem, I'm not going to say anything for now. Our growing season is winding down here anyway. For now, as you can see in the photo, I was at least able to yield some pretty nice hydrangeas after all and even this late in the season. However, I am guessing this story will be continued some time in the future. Thank you to all who gave their advice to help me here!