You say within a week if they accept it. What if they don't accept it? Will they proactively contact me to say so within the week?
If they don't accept it I honestly do not know what I can do next. The letter I submitted included a covering letter from an immigration attorney so hopefully that will also hold some weight.
The information on the page you shared is incomplete in that it tells you how to obtain one (roughly), but does not tell you what the eligibility criteria are. When you look on the Qatar ministry of Interior website it gives you the criteria - which I do not meet (having never been a resident).
I did attempt to obtain one anyway following the procedure, I enclosed all the things required, i.e. a letter asking for the PCC, details of my entries and exits, copies of my passport from the time, fingerprints from a UK police station, a return envelope, and the fee. The letter was returned to me unopened! But as I've said, I don't meet the criteria anyway so even if they had opened the letter I suspect it would have been ignored.
At the interview I presented all this information, including the returned unopened letter which had all the postage stamps on it, and the receipt showing I genuinely paid for the postage. The agent at the interview simply said, its strange they wouldn't open it. I couldn't disagree with him.