The thin plate and 100 degree angle countersink and flatheads would make total sense in this application, but looking at the pictures earlier in this thread, the countersink holes have a small diameter at the top surface. That makes me think of smaller angle like a 60 or even the 82. So I hadn't even considered that it would use a 100 degree flat head.
Looking at the 100 degree flathead screw dimensions on McMaster, they have the same OD as the standard 82 degree flathead socket head screws, so they may sit a bit closer to the surface, but they will still protrude above the surface since the head is larger than the large diameter of the countersink (I am taking this from the picture earlier in the thread that showed all the different flat heads, sae, metric and phillips, and all the heads were larger than the top large countersink diameter).
If they used a 100 degree countersink tool to match a 100 degree flathead screw properly during manufacturing then the countersunk hole should be flatter, shallower and the diameter at the top of the countersink should be larger than they appear to be.
So I forget that my brain is very good with spatial visualization in a way not everybody is. I agree at not throwing parts at it in a trial and error fashion, but I don't want to go through the geometry problem here unless you want me to. You can PM me and I can draw out the diagrams and all that or tell me to just post it here. I would be happy to help either way.
A set of digital calipers will help a lot to take the needed measurements. The cheapos from Harbor Freight work well enough and are reasonably priced if you don't already have a set. And since they are a tool that could help you buy less stuff in the future they will likely go over ok with your wife.
Also, (warning, a bit off topic rant) McMaster is awesome for finding stuff, part dimensions and if you use cad, getting files of nuts and bolts so you don't have to create them all yourself. My complaint is that they now seem to have a $10 minimum shipping and it seems to go quickly up from there as you add items. So when I need $4 dollars of bolts the shipping kills it. They are not usually the cheapest option for stuff either so where as I used to order metal stock, hardware and any specialty tools all from them, now I find it cheaper to get them from separate sources and can usually save $20-30 on a project. Doesn't seem like much but it adds up. Kind of a bummer.