It sounds like the camera might be shorting out the power supply. I vaguely remember having trouble with that exact model because the mounting holes are powered, so you must make sure the 4 metallic screws are electrically isolated from each other (so don't attach the camera to some solid metal piece; or use plastic screws).
It's also possible that the camera plus those two beefy-looking IR lights might be pushing the regulator beyond its limits even without a short. According to
https://raspi.tv/2017/how-much-power-does-pi-zero-w-use, the Pi Zero W draws 230mA when shooting a video with the standard camera. I imagine the IR camera without the lights draws a similar amount, and given that the regulator has to step up the voltage by a factor of about 2.0 (from 2 x 1.2V NiMH nominal voltage to 5V), the input current will be arount 500mA tops, well within acceptable limits. If we factor in the lights, however, all bets are off. Looking at
https://www.amazon.com/SUKRAGRAHA-Infrared-Illuminator-Adjustable-Raspberry/dp/B07RN44Z1Q/, the LEDs are said to be "3W" (not clear if each is 3W or they're 3W in total). Even assuming it's 3W total, that works out to an extra current consumption of 3W / 5V = 600mA, thus an extra 1.2A at the input for a total of 1.7A, waaaayyy beyond the maximum allowed by the regulator.
So, try dismounting the lights and see what happens. If the IR lights are the problem, you'll need a beefier regulator at the very least.