How to send a looped video to video feed?
Open Broadcaster Software (OBS) is a free and open source video suite that is capable of cutting from a live feed to a pre-recorded source video.
In additional to OBS, you will need obs-mac-virtualcam to map the output to a virtual camera:

Creates a virtual webcam device from the output of OBS. Especially useful for streaming smooth, composited video into Zoom, Hangouts, Jitsi etc. Like CatxFish/obs-virtualcam but for macOS.