Needed to launch the native application with:
cd <path_to_beamng>/BinLinux
PRESSURE_VESSEL_FILESYSTEMS_RW=$XDG_RUNTIME_DIR/wivrn/comp_ipc:/var/lib/flatpak/app/io.github.wivrn.wivrn OXR_PARALLEL_VIEWS=1 ./BeamNG.drive.x64
Initially, things took some time to load. On subsequent starts, it went much smoother and faster.
One thing I've found is that enabling VR in the menu at startup can cause issues and extended loading times; loading the track first then switching works much more reliably.