I don't think you can expedite either. The process goes as fast as it wants to go.

It's interesting that you got to keep your passport through the process. Mine got surrendered to HR the first day on arrival. I won't get it back until I either get kicked out of Qatar due to no medical clearance (should hear forthwith on that inshallah) or I get my RP. I carry a photocopy of my photo page, my arrival stamp, my permanent visa, and a business card with my HR Immigration rep's contact information. Does all this bother me? Not in the least. *shrug*