Hey, so I'm back with some unfortunate news. AMD's proprietary drivers would crash my system when running clinfo with just about every configuration, so I eventually found the open source OpenCL drivers and installed those. Everything is working great now, but Blender still does not recognize my card as a compute device.

Thanks, I'll reconfigure things to get OpenCL up and running properly. I wasn't able to find a way to do it on my own so if this works you have saved anyone else who buys a 390 for OpenCL and checks the Blender bugtracker a great deal of frustration.