(1) You don't need the stamped log book for the interview. That is for the CAA conversion.
(2) There are many FAA pilots at QR who made it through the process, probably without stamped logbooks.
So ask about this at the interview.
What I am saying is that if you can show them that you are being proactive and have started completing the FCL 050 logbook, and gathering supporting letters/ documents, it would help to make a positive impression and show that you really want to be there. Many pilots go abroad with a "why do you do it this way?" attitude.
All that will get you is a diversion direct to the exit fix, as that is one of the pilot attributes that they are screening for.
Maybe instead of "we don't do it that way in the USA" you might like to say something like "this is all that I am able to obtain, what have other pilots in my position done, and what do you recommend?"
In my case, as I said, they asked for letters on company letterhead from the Chief Pilot or fleet captain, which I was able to obtain. If you can obtain those now, and take them with you, that would be very helpful, I think. The one answer you are not going to get is "well, if you can't get stamps, then we will just take your word for it." You will have to do something to validate your flight time. That's not in QR's hands, it is in the government bureaucracy's domain, and they modelled their world on the UK, which is famous for paperwork and red tape.
As for the PRIA, I have worked for 3 different airlines that are now out of business. I can no longer get my training records from any of them. Airlines come and go. So just as a general practice, when you resign from a flight position, I recommend that you gather ALL the records you can, while you can. We never know in this biz.
Best of luck, and please keep us posted on your interview progress...
