That should work. I'd plant it near that fence and somewhat to the south of that window. Hopefully it is open ground on your neighbor's side of the fence and not a concrete slab.

Here is a link that might be useful: Palo Blanco

One issue you may want to consider is the width of the tree. It can be trimmed to be a bit more narrow, but the space will probably be pretty. If you use a taller tree the idea is to get the canopy above the house.

But for smaller deals and fast growers I like Yellow mexican bird of paradise, Caesalpinia mexicana. I grew one from seed outside a south facing window and it got to 15' and full in about 4 years. I watered daily for the first few years. Takes a trimming pretty well as long as the trunk is shaded which sounds like it would in your case. It gets a little taller

Palo Blanco is absolutely one of the loveliest desert plants on the continent, and a grossly underused landscape plant, IMHO. Despite its delicate appearance, it is windproof, sun-proof, and pretty much drought-proof. It will provide more ambiance than shade, however.