Welcome to the Forum, @ajbeck33! 
Thank you for writing such a detailed description of your experience! As I was reading it, I began to think about the timing of recent firmware updates and wondered if something hosed your camera and @Greeeen’s. Then I got toward the end and saw that you were also headed in that direction. 
This is what I think, yes.
I can’t say with certainty that this would be the case for that particular camera model, but I know that other Wyze Cams do write a log subdirectory in the root of the card, so I would expect to see the same with Cam Floodlight v2, and you’ll likely find multiple files there if you decide to pull it.
Also, I think that “click-click” is the camera’s IR filter cycling during startup.
I really wanted to suggest flashing your firmware from microSD with an older firmware version, but it looks like Wyze doesn’t make those available or allow that camera to do that, anyway, so this is my next thought:
Since that camera model doesn’t have Bluetooth or require you to scan a QR code but instead uses “access point setup”, I would consider taking it off the wall and temporarily moving it inside so it’s near your main wireless access point. If you have a strong Wi-Fi signal at the floodlight’s location then you can try leaving it there; if you decide to temporarily remove the floodlight, then you can power it indoors using an old extension cord or something similar. I would also temporarily disable mobile data on the phone being used to set it up and turn off every Wi-Fi radio on your gateway/router/access point(s) except for the 2.4 GHz band on your main wireless access point. The idea is to force your phone to connect to the 2.4 GHz SSID and maximize the signal strength to the floodlight in order to improve its chances of successfully getting the hand-off from your phone and binding to your LAN. I’ve had to take similar steps with other stubborn IoT devices in the past, including an original Wyze Plug last year. If you can get the floodlight connected to your home Wi-Fi that way, then you can turn the other things (5 GHz SSID, phone’s mobile data, etc.) back on and then re-mount the floodlight outdoors (if necessary).
It’s a huge pain in the bum, but I’m not sure what else to suggest at this point. Since this has happened to more than one person following this recent firmware update, I’m going to ask for some other eyes on this, too, and I’ll report back here if I learn anything.
Thank you both for reporting this!