As far as I'm aware there is no rank on the hat. We have my grandfathers Kalpak. He was a major in the TJFF 3rd company, calvary, stationed in Jisr al Majamie between 1945-1948. I have lots of photos of the unit but here is one of the officers. My grandfather is the 3rd from the right.
Unfortunately, I can't find pictures of the hat other than that of it damaged before we had it professionally repaired but seen as it shows exactly the same pattern that may already be of help to you.
I hope this helps,
D
