Like all things in Doha (you will quickly learn this) you can find the car seat you want one day and the next day it is no where to be found in any store. Stores don't stock things consistently and you cannot special order things usually. My baby recently outgrew her infant seat so I bought a new one for my 2 year old (goes up to 11 years!) and gave the baby her old one. I knew what I wanted and couldn't find it, and was told it wasn't in Doha. So, I bought one that was fine but only went up to 4 years old. Anyway, fortunately for me, the store guy didn't know what he was talking about and I found the one I wanted at the same store, different branch within the week and made an exchange.
So, short story long... Bring your car seats. You don't want to have to rely on the stores stocking them at exact time you move here. They might, then maybe they won't. You can never know.