Jeremy,
You are right about the black version being different from the white version (The Hero3+: https://gopro.com/support/articles/hero3plus-camera-comparison, and the Hero3: http://www.unsponsored.co.uk/press/gopro-hero-3-black-vs-silver-vs-white/). I cannot say this with 100% certainty, but can say with 98% certainty that you will run into problems trying to swap the motherboard. The specs are quite different between the two, which means the hardware between the two are substantially different. This usually means that a simple mix and match of hardware will not work, mainly due to the firmware protocols implanted into these devices.
But as always, when there is a will there is a way. I just imagine it will need a lot of willing.
Hope this helps!
Scott