Is have this chair from costco but it's not there so I found it on Amazon. chair
Well, my wife is 5'2" and seems to be satisfied with....
https://www.amazon.com/Bayside-0764053498039-Furniture-Seat-Chair/dp/B011XMS0ZA/
I believe this is what I'm sitting in at the moment. And hey, at $197 it's almost an exact fit for your budget.
https://imgur.com/gallery/ir0TI
Here is the tube I'm looking for the type of plastic if it's available but I started drilling and for what it's worth it's pretty hard to drill into haha
edit: heres the chair in amazon:https://www.amazon.com/Metrex-II-Mesh-Task-Chair/dp/B011XMS0ZA/ref=sr_1_3?ie=UTF8&qid=1483204954&sr=8-3&keywords=BAYSIDE+FURNISHINGS
edit: cant seem to find type of plastic, im is there like a standard plastic used in normal office chairs?