could you add a picture that shows the area from outside of the screen enclosure?

sure, here you go

Possible choices, you would have to research these further:
Fragrant tea olive, mockorange, bamboo (non-invasive types), Little Gem magnolia. I was looking for something that would be evergreen (since your zone allows you year-round use of your area) and fairly narrow.
Laurie

Oh also possibly camellias.

We looked into Bamboo, specifically Graceful. But my wife had some reservations, thinking it will grow out too much and still be too tall for the area. I'll do some research on the others you mentioned. Thanks!

I would place the palms elsewhere. Also recommend you have the flower stalks trimmed off every year right after bloom because the ripened fruit makes a stinking g-dawful mess.

I think you would be far better off putting a fabric screening material that is used on greenhouses or a window film on part of your glass roofing to create a privacy screen rather than using plants. You don't have a lot of room there and maintenance will always be an issue. Bamboo is inherently messy and is not "low maintenance".

Thanks for your input. We are also considering some sort of outdoor blinds for the vertical portion of the screen. Window film isn't an option because everything is screen, no glass.

Rhapis palm would do it but require a root barrier so as not to invade the neighbor's yard. More conventional solution: Viburnum odoratissimum. Love tea olive & camellia, but have yet to have luck with these plants here. Not sure why ... the sandy soil my require large amounts of organic matter to be added.