There are two kinds of "HDMI splitter". There is the switched type, that lets you have one active device at a time from the one port on the desktop HDMI port.
The whole signal goes to that active device.
Then there is the ordinary "splitter" that gives half a signal to each of two devices for same image on both. There might be a slightly lower signal strength, but might not be serious problem.
Your video system may have another port, like DVI or VGA. There are electronic adapters for VGA to HDMI, and DVI to HDMI cables you can use for the monitor. No sound, but no problem since most monitors have no speakers.
Another choice is to add a video port to your desktop. There are fine USB to HDMI external video adapters, not expensive. Put your monitor on that, and plug in the VR headset to the laptop.
I have used both types of HDMI splitter. I have used external video adapters for USB, and different types of cable adapters.