Right after I posted, I saw a solution Ryan had that used a spring under each corner of the bed, with a screw pulling it down to the spoil board, which seems like an easy enough solution.
You don’t have to move the bed to have auto leveling. If you can measure the Z height at a couple of places, then the software can move the printer head up and down to get the correct distance to the bed at each X,Y location. It’s also called auto tramming. I think marlin and repetier firmwares are starting to support it, but it’s a lot of code, so it may not be enabled by default (I’ve never tried it).
For the record, I printed all mine the same way, without thinking about it, and it’s OK, just not great. There are some plated versions of the cable chain that have the parts rotated 90 degrees four times, so parts pointing in each NSEW (That’s north, south east, west, and completely suitable for work). I can’t think of a way that you could skew the parts to make it necessary to need all 4 directions, I think just NS would be fine. Certainly a big improvement. Just print the other direction ones in a different color, and make a zebra pattern. 🙂