I finally managed to get all three of my monitors working with my Radeon R9 280X (model ASUS-R9280X-DC2T-3GD5). Previously, using more than 2 monitors would result in my system freezing until I unplugged one of the outputs. After trying patched framebuffers and a variety of different port configurations I finally solved my problem!
It turns out the solution was to switch to using the card in UEFI GOP mode. To accomplish this you need to be using Clover in UEFI mode (not CloverEFI), and your motherboard needs to be set so that the GPU runs in UEFI GOP mode. On my motherboard this meant disabling CSM.
My system now runs great with all three monitors, either using GraphicsEnabler=No or GraphicsEnabler=Yes along with a patched framebuffer. This may also work for other AMD cards that exhibit similar behaviour when used with three monitors.