For anyone who's struggles with the setup quest like I do, with the new scope eye-relief changes, you can run viably scope on shotguns! If you've seen Deadly Slob's Monster 153 that always comes back on insurance due to it's huge size, I made a similar 153 to do this quest, and got 3/5 back on insurance.
I tried using the 133 to see if it had any better accuracy at full build, but it had the same accuracy, and because it's not a bolt action, you can't hold the fire button to wait to rack the action until you see if you shot landed. Definitely go with the 153 for the ability to do follow up shots. I did not try any of the other shotguns since the quest requires 133/153, however I did an unrelated SAIGA-12 Sniping build, and was only able to get to something like 8-12 MOA.
THE BUILD: No pic since I'm at work, but I'll try to add a picture of the build when I get home tonight.
Using a 750mm barrel, 8 rnd magazine, and the pistol grip/butstock combo I used an MOE + Buttpad, and the SPRM Mount from Jaeger. Finally for the muzzle, the Remington Choke and Salvo-12 give the best accuracy, but the hexagon suppressor is also useable. With AP-20 Slugs and the hexagon suppressor, I was getting a 4.85 MOA accuracy, which made sniping absolutely viable. There are other more accurate Slugs, but I went with AP-20 for the potential to 1 tap to the thorax. I personally used the ELCAN or Tan ACOG + RMR for a cheap ranged optic that was useable at close range as well, and headed to my normal Shooter Born in Heaven spots (not telling sorry :p).
AIMING/ZEROING: The slug seemed to hit high consistently, so the best method I found was to leave the zeroing at 50m, and aim shoulder height for anything 100m or closer, and seemed to be dead on at 150m. This took some experimentation, and many shots over the head when I zero'd to the correct distance. With the poor MOA accuracy compared to rifles/bolt actions, it may take a 2nd shot to hit where you were aiming, but I was consistently dropping PMC's from 100-200m. With AP-20, you can also 1 shot to the thorax through level 3 armor, so hitting a headshot isn't SUPER important, and I was usually able to watch an enemy for long enough to tell if they had armor I could pen. I was never able to get a feel for the travel time of the slug, and only hit maybe 10 of the 30 shots I took at running targets at long range not getting a single kill of these hits. I'd recommend waiting until the target stops.
I Sniped with this setup for all 15 kills, and ended up getting 8 of them as 100m+ headshots, so SBIH on customs is done for me, and I'm honestly considering trying to use it for the rest of my quest, however on a map like reserve with 300m+ shots, at that range this shotgun wouldn't be reliably accurate.
Unfortunately I didn't take any clips of my kills, but there were a couple of gnarly shots. If there's interest, I'll try to use if for my other SBIH shots and take some clips to showcase what this build can do.