Since the contest requires it, and since I finally know what shape this badge is going to have, here's a little mockup image, at 1:1 scale, with some of the parts placed next to it to give a better idea of how it's going to look like.

I'm considering making it so that you can break off the bottom part and wear it as a badge alone, and use the electronic part of the badge separately. The battery holder is too big, because it's a 4 AAA one, not 3 AAA — I didn't have the correct size at hand.