I think this is the design that’s going to work. I found side-mounted LEDs on Digikey that are pretty cheap as well as a very minimalistic battery clip that is surface-mounted.
The etch was very clean:
I also bought some DIY solder mask off of eBay. Seeedstudio used to sell it too, but stopped carrying it. We’ll see how it works; I have high hopes for it though. If I can get it to work, then the solder paste reflow technique may work really well for this design.