Well HELL, it what I thought when I first read this...as I am in almost the same boat. I am flying Qatar to Johannesburg in mid-May. Their website says to call 96 hours ahead of departure to get pre-auth for firearms or dangerous items. So seeing this thread and feeling dread, I called them.
I did not receive a totally consistent response, but I may end up in a slightly different situation too, since I booked the trip through Alaska Air. I live in Soldotna, so have to fly out of Anchorage. Unless I am flying Delta, my initial leg with almost always with Alaska Air, and we fly with them often throughout the year. They are super easy on firearms and ammo, so love to use them when I can. (Note for Domestic Alaska flights flights, we can fly with up to 50lbs of ammunition per passenger, in factory boxes or non-crushable containers...way easier then trying to ship it up).
I had a choice of booking through Qatar with an Alaska flight between Anchorage and Seattle on each end of the trip, or booking directly with Qatar with the exact same flights, so a mixed itinerary regardless. It was a couple hundred dollars more directly through Qatar, so I booked it through Alaska hoping I would get bonus miles on my elite status. So my actual booking is through Alaska Air, but all international flights with Qatar.
So I called Qatar asking about pre-approval for my firearm, letting them know I had the forms all prepared and ready. Also asked (having read above) if I should only check my rifle through Seattle so I can recheck it with Qatar after landing, even re-do the TSA portion if necessary.
Was told I do not need pre-approval through them at all. Alaska Air will handle checking in the firearm and transfer all information and such along to them. That I did not need to check just to Seattle, that I could and should check it all the way to Joburg.
Certainly liked this answer better than so sorry, cannot accommodate your rifle, but also not overly relieved by the inconsistency in the answer. I almost asked what the policy would have been for this exact same itinerary if I had booked with through them vs. Alaska Air. Would seem strange that the exact same flights with the same airlines, but booked with one vs. the other could have opposite outcomes, but who am I to say.