Okay, here's a followup. I swapped out the AX860i with an AX860 (no "i"), figuring it most likely wouldn't work. It did. First boot gave me the "catastrophic" event warning, but I think that was left over from the first power supply. I have now been using my machine for 24 hours, with numerous warm and cold boots, and I have not had the catastrophic event warning once. I was also able to update the BIOS.
There is still an issue, however. The problem of the LAN1 port not functioning after a cold boot remains. A restart from Windows fixes it. LAN2 does not exhibit this problem, and since I have never needed more than 1 LAN port, I have opted to just keep my cable connected to LAN2.
I don't know if these issues are limited to Corsair power supplies, but it's something Intel needs to look into. With the popularity of the AX860i, they're going to have a lot of unhappy customers.