If the buttons are supported, but still don't work, then the next thing I would try is to write a config file to place in /etc/X11/xorg.conf.d/. Haven't done that for years so I'd have to research it again.
If all fails, then you could try using emulation. You could check in the Xorg.0.log to see if libinput is the current input driver:
grep -i driver /var/log/Xorg.0.log
If it is then you can use the config:
Option "MiddleEmulation" "bool"
Enables middle button emulation. When enabled, pressing the left and right buttons simultane‐
ously produces a middle mouse button click.
so that you can use the two working buttons to emulate the middle button. Check out man libinput. You would have to write a suitable config file for /etc/X11/xorg.conf/ with this variable, and hope that X honours it.
Make sure xorg-x11-drv-libinput, and probably xorg-x11-drv-evdev are installed, though I think the mouse used libinput. Perhaps re-install them.