The AAA battery holders finally arrived, so I could complete the prototype:

I also got to experiment with adding some wooden parts to the bottom, to make the whole thing a bit easier to hold. For now I cut them out of wood by hand, but ultimately they would be laser-cut out of acrylic:

The whole thing fits very well with the mini-breadboard boards:

I will need to make a sticker with pin labels for the top of the display, so it's easier to see what is what.

Finally, I got the first responses from the fabs, and it seems that at small quantity (50 pieces) the price will still be relatively high, around $9, but at larger quantities (300 pieces) it can get as low as $6 (that's without the shipping costs and margins, of course). This is for a partially assembled kit, where you need to solder the matrix and the battery holder yourself.