OK, latest results:
1. Added more insulation to the left hand side panel of the inner chamber. Did a "up to 250° C" ramp-up / ramp-down test from a cold start and found that it took ~230 seconds to reach 250°. That is consistent with my earlier observations when doing it with the oven cold. By and large, adding insulation to the inner walls of the cooking chamber did not seem to have a ton of impact on the "time to reach 250°" performance, but it
did have a noticeable impact on keeping that side chamber cooler. So net-net, adding insulation there seems to be a win and a generally desirable thing.
2. Suspecting that the glass panel on the front door was a significant source of heat loss, but not wanting to permanently block my view into the oven, I made two slip-in heat shields out of two of those pieces of aluminum flashing and some of the insulation. The flashing pieces coincidentally turned out to be exactly the right height, so no cutting was necessary. And there's a little ridge near the bottom of the door that the heat shield will catch on, which helps keep it in place while shutting the door. Of course I could bodge in some Kapton tape to hold them in place if I wanted to.
These pieces just lie inside the door frame and are held in place by the door itself.
Reflecty stuff, cool.
The way they lean up against the frame. When the door is shut, they're locked into place well enough.
View from outside with the shields in place, and the door closed.
Did another 250° ramp-up / ramp-down test from cold start, and found that adding those brought the "time to 250°" performance down to 196 seconds. So yeah, it appears that the glass front is a major source of heat loss. Closing down that channel even partially helps by a decent amount.
Now to finish up that control box and get some proper software written for this thing.
On the software side, here's what I think I'm going to try for a first, "strawman" stab at this. Instead of dealing with PID and complicated algorithms, I'm going to take the data from one of my test runs, run linear regression on the side of the curve up to 250°, and use the slope as an approximation that's good enough to use heuristically. Then I'll do something similar to what user @fourfathom was doing, where I'll just rely on timing for the various states, based on the expected change in temp over time. I'm just going to hard-code the timings into my control program for a trial run, and test that approach and see how close I can get to matching a published "official" soldering profile.
I also just ordered a few more of those SOIC-14 to DIP adapter boards and 30 more NE556 timer chips in SOIC-14 format, along with a fresh tube of Kester 63/37 solder paste, so I can start experimenting on actual boards soon.