First, the things that went wrong.
As the primary goal of this project was to test the slotting tolerances and jointing with solder pads and via, I didn't put much (any?) though into the circuit. I used an LR44 battery because I'd just bought like 100 after my calipers died. Welp, an LR44 isn't powerful enough to light one, let alone two LEDs in series. But!!! With a different power source the circuit did in fact work!
I have a more complicated design in the works and something I realized early on is I needed good naming for all the parts to maintain the pathways in my head.
This design contains 3 main parts; Base board, Arm board, LED board. Interconnects get abbreviations like; B2A/A2B. Where B2A is on the Base board for the connection to the Arm Board. A2B is the corresponding connection on the Arm Board.