Won't boot? Green light with usb charger

[Flavor]

You are correct. The Freeplay CM3 requires a battery to run. It will randomly shut itself down if you don't have one installed.

Do you think that you have the SD card sorted out now?

From our records, it seems that you have a 2pack of batteries. Here's a test for you.

Disconnect both batteries. Plug in power to the MicroUSB. The back LED should go green. Plug in one battery to one battery socket. The LED should turn blue (unless the battery is already 100% charged).
Disconnect that battery, and it should go green again. Plug in the other battery to the other battery socket. The LED should go blue again.
Connect both batteries to both sockets. The LED should stay blue. You can then let it charge until it is green.
[You could also fully charge one battery and then the other.]

This test will make sure both batteries are working.

I don't really know what's up with your power button. I guess there's a chance it was soldered at a bit of an angle or something. Early on in our first prototypes (when we called it the Game Pie Advance), there were a couple people that had to glue some bits to the slider to make sure it engaged well. That's when we were soldering by hand, so inconsistencies were more likely. It's possible that something like that happened.