I have an update, and a solution! I dug around some more, retraced the steps I followed to get it working on Arch, and found out that OpenMandriva, unlike most other desktop-oriented distros, doesn’t enable Avahi (zeroconf) by default. I got it to work by running this:
# systemctl enable avahi-daemon.service
and rebooting and trying it again.
Thanks to everybody who tried to help! I’ve never used the Brother driver package on any distro, and using Avahi doesn’t require the printer to have a static IP or installation of any drivers other than what CUPS provides. Avahi basically enables CUPS to pick up driverless wireless (IPP Everywhere) printers automatically.
And to answer the questions about desktop environment, I’ve been primarily using KDE for many years, which is one thing that drew me to OpenMandriva