I figured this out and got it working this morning. Posting the solution here in case it helps someone else.
The problem was that UEFI Secure Boot was blocking the driver from being loaded by the kernel. I didn't read the secure boot instructions carefully enough. During the install of amdgpu-pro you are asked to disable secure boot and then asked to supply a password. Then on reboot you are confronted with a boot screen that looks like it's asking for your password. In reality it is asking you to type certain characters of your password, one at a time. See below for some more information, apparently this happens to a lot of people:
Once I figure out how to disable secure boot, then the card and driver now work together very well.