The joke's on me. I've been saying all along that the vents in my house are 12x4 inches. That's true, and I've taken the measurement a couple of times to make sure I didn't fake myself out. Now that the assemblies of fans are ready, it was time to put the first one into a vent cavity. That's a different one from where I have been measuring. It's still 12x4, but there were several irregular protrusions where pieces of the hardwood floor had been cut to make the opening for the vent. There was enough irregularity that it was pretty difficult to get the fan assembly into the vent cavity and then put the register in place between the risers of the fan assembly.

The bracket design for the fans was done parametrically in OpenSCAD, so it was straightforward to make dimensional adjustments. I did have to change my philosophy about it, though. Here's my reference image of the bracket design (the dimensions are intentionally absurd so that the text labels are in the right places and legible).

Previously, I had been thinking that I wanted to risers on the bracket to be as close to the walls of the vent cavity as possible, so the register would sit loosely between them. The weight of the register would hold the horizontal tabs in place. My change in thinking is that the riser horizontal distance is now pretty close to the fan major size (in fact, identical). The register now pushes the top of the risers outward while continuing to press down on the horizontal tabs. I printed the brackets in PETG, which has reasonable flexibility, as you can see in the professional demonstration in this picture:

When you get right down to it, I don't have a tremendous amount of slack anyhow. I'm using 92mm fans. WIth the overhead of the square case holding the fan, it's about 95mm. The thickness that I arbitrarily chose for the thickness of the bracket band is 2mm, giving a total of 99mm (about 3.9 inches). So, yeah: tight.

Things were also reasonably snug in the long dimension, too. I originally used the same brackets on the end fans as I did for connecting two fans side by side. That meant a half of the bracket band width sticking out from each end of the assembly. It still fit into the cavity, though. Since I was going to reprint the brackets I had already made, it was a simple matter to tweak the parameters in the design to make a half-width bracket with single holes for the ends. You can see a couple in the background of the image above.