Is it grammatically correct to say 西南人?
As a 西南人, I have to say that it is rare to hear. 南方人 (Southerner) is okay.

If someone is from the south side of the Huanghe River, they would let you know which province they are literally from, like Sichuan, Yunnan or Guizhou.

I suppose the reason is dialect, which so different in southern China.

But 西北人 (Northwestern) is commonly heard.