The PCBs with the horizontal layout and display module arrived, so I assembled the new version:
I'm happy with it mechanically. You don't touch the contacts when you hold it, and the two buttons on the micro:bit are right under your index fingers. It's wide enough to hold comfortably. The flattened d-pad is a bit less convenient than it could be if it was full-size, but I can't have everything. The coin battery fits inside the sandwich, and the whole thing the right thickness. I had to add some spacers to the bolts that hold the thing together, to make sure the spacing is right. You can both use through-hole bolts, or solder them to the pads — here I soldered them, but I think I will go with through-hole in the future.
I'm still missing one crucial part — the attiny that will be handling the buttons is still on the slow boat from China. Seems like the holiday season was really tough on the post offices around the world, because everything takes longer now. I guess I won't be able to get this to work before the contest deadline.
On an unrelated note, I think I figured out why I couldn't get that display to work reliably without a module. Turns out that after powering it, it requires to be reset by the physical reset pin. The datasheet's example schematics don't show it, but those modules have an R-C circuit on the reset pin to do that. Mine didn't, hence the problems. But I think I will stay with the modules for now anyways.