Evergreen, right?
I should think that the least aggressive vines would be a jasmine blended with perhaps bignonia. But, you're going to need an interior support of wire mesh and realize that what you plant is going to need annual trimming.

Wow, just wondering what 500 feet of cedar (open,splitrail?) fence costs nowadays?

bignonia is super invasive,,,its a wisteria cousin and acts much like it.

pyracantha would do well in southern texas as it tolerates heat well and has those beautiful orange berries. be advised its thorny tho

kaliaman is probably right about bignonia.
My frame of reference is zone 7-8 and much different from
South Texas.

I haven't found bignonia (crossvine) to be invasive. You may be thinking of the vine that resembles it and is horribly invasive, Campsis radicans (trumpet vine). Love my crossvine, which blooms like mad in the spring with sporadic rebloom.

What about bouganvillea?? Here it's an annual but hopefully not in South Texas. Showy, showy, showy!

Hope you'll let us know what you plant. Good luck!