My AX860i also had random shutdown problems, but I have finally figured out the issue. For me, it's because by default, the AX860i is set to operate in Multi Rail OCP (overcurrent protection). I believe the same is true for the AX1200i.
Select your PSU in the "Settings" category in iCUE to change this setting to Single OCP (or to higher per-rail shutdown thresholds). If you find that your PSU is by default configured to be in Multi Rail mode, this could be your problem.
From my understanding, what Multi Rail OCP means is that each CPU/GPU cable (and the SATA category as a whole?) that's plugged into the PSU will have their own power limit. If the hardware on any rail goes over that power consumption limit, it will cause the PSU to shut down.
What could be happening to you is that at a certain time of the day, your computer starts to do semi-scheduled things that increases the power consumption, which trips one or more of the rail limits in Multi Rail OCP.
I fixed my PSU's shutdown problems by changing the settings back to Single OCP mode, which means the PSU will shut down if the sum total of all hardware on your PSU (per voltage rail) exceeds a preset power limit (i.e., for the AX860i, 859W for the 12V pieces, 125W for the 5V pieces, 82.5W for the 3.3V pieces; for the AX1200i, 1204W for 12V, 150W for 5V, and 99W for 3.3V). You can also increase the per-rail threshold in Multi Rail, if you know for sure how much power the pieces on each rail should draw.
Corsair's choice to set the default to Multi Rail is probably a huge mistake in terms of user-friendliness. Ironically, the digital architecture (the "i" in AXi) is what allowed them to have this (admittedly quite advanced) feature in the first place. Reading around the internet I find that many people have random shutdown problems, and many of them have RMA'd their unit, but as far as I know they didn't test whether this setting is the cause.