The problem is on the right-hand side door, where there is only a middle lock. The piece that actually rotates when it hits the left-hand side and holds the right-hand side door closed, is now rotating freely, it is not attached anymore to the rest of the door lock mechanism. I will try and take it off the truck and take pictures soon.
I know some Suburban-specific parts are hard to find, that is why I was hoping another vehicle used that same door lock (hence it would be easier to find!).
I am adding this a few minutes later, it looks like Classic Industries has a repro of what I need:
https://www.classicindustries.com/product/1956/chevrolet/truck/parts/cx1277.html. I will have to take mine out to double-check, but that would be pretty cool!