Thank you guys for helping me out with this!
I think from what you guys are saying, I've figured out the problem. I'll attach a photo of the design (you guys are so going to laugh at me
). There are two pieces to it, one that nests on top of the other. They're held together with elastic bands on the sides. Actually fun fact: in one of the attached photos you can see what the cafe looked like where one of the images was captured, while it was happening.
To answer questions about mounting, the pinhole is attached with gaff tape to a piece of mat board, and the shutter is a piece of acrylic that slides over, revealing a hole. This acrylic is a little over 1/8" thick. I made very sure that all the openings behind the lens were large enough to not cause a vignette, but I wasn't careful enough with the front. The effect may come from light reflecting off that acrylic opening. When I use it in EV 13-14 where the shutter is only open for a minute or so, you can actually see it (also attached). I'm planning on using a countersink bit around the acrylic opening to solve that problem, so that might reduce the flare effect, too.
For those more curious about the design, here's a video rendering of the first attempt I did from Fusion 360, it shows how it works:
Also, the back of the copper used for the pinhole is reflective, though the rest of the inside is black mat board and is totally light proof. In another thread someone mentioned a blackening method I'm going to try, or I may try. My first thought when I saw it was that light might be reflecting off the paper back onto the copper, creating the effect. After thinking about it more the acrylic thing makes a lot of sense, but I should still try doing better about blackening the inside.