Is "9" your Sunset Zone? What type of fence/wall is it, and how high? What is the size and material are the planned container(s)?

It is the container that will make it trickier for plants to thrive in the hot conditions. Is there a reason that you cannot instead use wire between the attachments that would hold up the proposed string? I have both vines and espaliered Grewia occidentalis (Lavender Starflower) shrubs along 5' high and 7' high block walls, as well as along a section of our house exterior. I'm a big fan of the Grewia, for a whole host of reasons. We inserted some screws into the blocks/stucco, then used a strand of wire to span the distance between (3 strands for the 7' wall; the house wall Grewia has only one wire strand holding up the 10' espalier, and for the first 8' of growth it didn't even have that--it just leaned against the wall! lol). There are also attachments you can glue to walls that can support the strands.

Yellow catsclaw? Fast,tough,takes desert heat-frigid temps by California standards.

Thank you guys! We are looking into shade cloths, and we will probably put one up as well.

My instant go to would be some sort of wire, but I'm a) worried it would be too hot for the plants in the heat, and therefore fry the vines, and b) is a bit too permanent than the landlord will like.

The house is about 50 feet long and 20 high? Maybe? Can't measure. I'm thinking morning glory at this point, since it's not as top heavy as other vines.

And I've heard too many horror stories about cat's claw to go that route, although I think it's GORGEOUS.

I live inland, temperatures over 100, dry air. Sounds same as you. Growing anything in the sun in a container is very difficult and especially if the container is black. The soil in there gets too warm in the sun. In the earth the roots stay cooler. There is a container forum and they may tell you more about how large the container must be to stay cool.

Your choice of trellis eliminates most vines that get woody. Annual morning glory sounds best.

If you could plant in the ground I would suggest Bougainvillea.

I thought you meant shading a fence type wall to block the reflected heat onto the house. A vine on the house wall is not your best solution IMO. Too many unanswered questions to make recommendations though.