Is ".... au sud-ouest" incorrect here?
Yes, in that case. You would say au sud-ouest when talking about a relative position.
Ils ont une grande maison au sud-ouest de la ville (which means outside the town.)
Yes, in English also that would be "to the southwest" (of another location). So when it comes to translation
dans le sud-ouest => in the southwest
au sud-ouest (d'une autre location) => to the southwest (of another location)
And the two are NOT interchangeable!
Thanks, Jojo, that is very clear and helpful.