I’ve since found my own solution.
If anyone else has the same problem, here’s how I solved it.
Handbrake requires the driver amf-amdgpu-pro
LINK: AUR (en) - amdgpu-pro-installer
Use the following commands in the terminal:
git clone https://aur.archlinux.org/amdgpu-pro-installer.git
cd amdgpu-pro-installer
makepkg --syncdeps
makepkg --install
Then restart the system.
AMD VCE is now available in Handbrake.