So, if I understand you correctly, you're saying that you have an 18" wide space with a 4' tall chain link on one side and a 6' tall (solid?) fence on the other...right?

If you plant bamboo in that space, how will you be able to get in there to do any maintenance, like cutting out dead culms? And what kind of sun will the bamboo get? In addition, if you put in a clumping bamboo, what happens when the diameter of the clump starts exceeding the 18" space? I am a little dubious about this as a longterm solution.
Would it be possible for you to plant it on the inside of your new fence, rather than between the two?

You are correct. It is a very narrow space. probably too narrow.

I wish I could plant it on the inside of the privacy fence, but I have 6 established 8 foot tall queen palms and a stone planter that I would have to move out. I had arrecas in between the queen palms, hoping they would grow into a large screen. The cold really shocked them terribly and I think it will take years for them to recover.

I was hoping there would be enough room in between the fences, but he more I look at the space, the more I think I would be creating a mess. I will try experimenting with the bamboo in other areas.

The only other thing I can suggest is putting bamboo in pots so that when you need to maintain the bamboo you could move the pots out to work on the bamboo.