One paint job and 4 hours of paint thinner fume exposure later, I have the minigun completed. Total time: a little over a week from start to finish.
The Team Fortress 2 minigun, or "Sasha" as it's referred to by the Heavy Weapons Guy, is the most pain in the ass object to make in terms of weapons. The flamethrower had nowhere near as many components as this had. The sheer weight of Sasha rivals that of anything a 12" figure is capable of supporting without forcibly moving their limbs or tipping it over. I'll likely need a special stand for assisting Sasha in poses.
I used a bottle of Flat Black Testors Enamel paint that I had lying around for about four years. Still decent! The quality was a bit thicker than I'd like, and a bit of thinner was needed. The Gunmetal Grey was a bit better despite also being unused for four years. I used some of the Aluminum paint for the silvery part. A recommendation to anyone who decides to try building something like this: paint your parts first. There were way too many small crevices that required a bit of effort to paint when most of it was built.
Here's a sketch of the front profile of the minigun, as traced from the Valve poster. A very handy image to have to work with. I'll finish off with a few more shots of the minigun from different angles to get a sense of the details.