I'm going to order a new version of the spacer, driver, and base board. Here's what's changed:

1. Thicker copper. I'm going with Oshpark's 2oz copper option. This means the boards themselves will be 0.8mm, but that's fine.
2. Less soldermask. I've taken away the soldermask wherever the different PCBs come into contact with each other. I have no idea if this will help, but it can't hurt.
3. More careful routing. There's a chance that the copper traces were creating high points and preventing contact on the prior revision. And I had to move most of the traces anyway, since I got rid of the soldermask.
4. Better baseboard. Now it'll be easier to grip the base PCB and not have my fingers get in the way.
5. "SMD" LED. On the driver PCB, I'm doing everything I can to keep the bottom side as flat as possible. That means no through-hole components. But I still want a through-hole LED as the "pin" on the driving wheel. So I'll fold the LED's legs over and solder it like a surface-mount part.

Here's what hasn't changed:

1. The Geneva wheel itself. I'll still be using the same ones from the first batch of PCBs. No need to order again.

I haven't ordered yet, so it's not too late to suggest other changes or give advice on this version!