Nick,it almost looks to me like those studs are turned around in the wrong direction.
I may be wrong,but the ends with the short threads go down and thread into the aluminum sprocket hub;the ends with the long threads are facing up toward the sprocket. I would consider getting new OEM sprocket studs along with nuts.
I'll bet if he puts the sprocket on, the threads are below the sprocket surface
The sprocket has a difficult time squeezing onto the shouldered parts of the studs.
I was not able to get the new sprocket onto the studs just as you suggested.
First off, welcome to the forum, we're glad to have ya! Second, thank you for your service - my entire family is military (mom did 20 years in the Navy [16 active, 4 reserve], dad did 8 years in the Corps [sniper in desert storm] - now a E9 in the Army Reserves here in Michigan, step-dad was a medic in Vietnam for the Army, grandpa died in Vietnam months before it actually started as the "Vietnam War", most of my friends went in straight after high school, ect.
Third, I love the story and history behind your old man's K8 - too bad you can't track it down and get it back.
The K7/8s are awesome in my personal opinion, but perhaps I'm a bit bias!
Anyways, the sprocket should rest just like in the photo. I had initially put mine in backwards and noticed something didn't look quite right. I will say that removing the studs that were in there was an absolute nightmare. I ended up welding the nut to the stud and then torqueing it out with some heat!