The Linx Toolkit for Raspberry Pi supports SPI but it only makes spidev0.1 available as SPI device 1 but not the spidev0.0. Adding that in the liblinxdevice shared library is a trivial fix but of course requires GCC recompilation of the shared library.
As to the Raspberry Zero, no there is no chance that this gets supported anytime soon as an embedded target where you can directly deploy LabVIEW VIs onto. It uses the BCM2835 CPU which has the ARMv6 architecture but the LabVIEW compiled code requires at least an ARMv7 architecture, which the BCM2836 on the Raspberry Bi 2 Model B has (and all newer models after the Pi 2 have either a BCM2837 or BCM2711 CPU which also support the ARMv7 architecture).
I'm entertaining however the idea to create a Linx server from the existing code that could be recompiled for the Raspberry Pi Zero (and would most likely run on the older Raspberry Pi A and B too). In that way you could communicate to the Pi Zero in the same way as you can now to an Arduino and other such Linx supported lower end embedded controllers over a serial port or TCP/IP.