Behind the tree and in front of that chain link fence? About how big is that spot? Looks like root competition might be an issue.

Yes, between the tree and the fence. It's about 3 feet between the tree to the fence, and about 6 feet wide.

Firebush (Hamelia patens) will grow in partial shade and has butterfly attracting flowers...

https://www.fnps.org/plant/hamelia-patens

IMPE, it gets nice and bushy when regularly clipped.

If you don't have luck with a plant, I'd look at some bamboo or reed flexible fencing. It can easily be attached to the cyclone fence and in the shadows back there, would barely be visible. Some aren't opaque, but I've put some in similar places and have just doubled them over for more privacy.
These beefier versions have more privacy

Gold dust plants

Geographical location of planting site?

Here are some suggestions using aucuba crotonifolia & clivia:

Thank you all for these great suggestions! I really like the bamboo fencing idea so I don't have to contend with the existing root system.