So using paper does work, but it is sort of hard to do. I don't know how long this will last before one of the mounts lifts out.
I found 2 techniques that work. The pictures are for the first technique. The second technique is similar to the first, except you use less paper, going only around the middle (the ends aren't covered). This wedges the mount by the sides only, but seems to be enough to keep it in place. I found this easier than the technique in the pictures, but I didn't figure it out until the last key

.
Here is the plate mount stabilizer bracket. The notch on the upper left is where you clip the bar onto, and that is what I am worried will snap eventually. The wedge thing on the bottom left is where this bracket would clip into the plate. The Ergodox doesn't have anything for this guy to latch into, which is the heart of the problem.

The full assembly for the MX Cherry stabilizer. Consists of 2 brackets and one stabilizer bar.

And for completeness, here is the 2x key with the bar inserts. One side is slightly longer than the other (in the pic, the right side), and it does matter which way you put it on. Otherwise, the key won't go down properly. It is also kind of hard to put on, especially when the mounts aren't terribly secure.

The case designer had the right idea, but the wells are smooth all the way down to the PCB, so there isn't anything to latch onto or clip into.

But take a few strips of paper and lay them over the slots.

Push the assembly through

Rip off the excess

And now you have something that should keep the brackets in place. It may take a few tries to get something that actually holds, especially if too much paper gets ripped off when you are removing the extra.
Also, I've found that a sheet of paper folded once was enough to keep things tight. Unfolded, the sheet was too loose, and folded twice, it was too difficult to fit into the slot.
Anyway, there is my shoddy solution to this problem.