Thanks for the detailed response. I have some updates.
I had taken the monitor off my desk since I could not even get it to start up anymore. I finally got around to looking at it today.
I plugged the monitor in to start troubleshooting and it started right up

I tore the LCD panel assembly apart to see what the LED configuration was. See the attached picture. There are 2 strings of 16 LEDs along one side. I applied voltage and measured the current of both strings. At 50V, one string draws ~45mA, the other draws ~60mA. I would expect them to be equal. So it seems one string is starting to die.
I also found a schematic of a similar monitor, but it seems to match my board as well. (Attached).
On the backlight connector pins 1 and 2 are shorted together as well as 5 and 6. Seems they have a generic driver circuit for up to 4 strings of LEDs and they just tie 2 pairs together for the 2 string monitors.
Probing FB1 gives the following:

Probing FB4 gives the following:

There is a clear difference between the two.
An even bigger difference is seen between VFB1 and VFB4 which are inputs to the main processor:
VFB1:

VFB4:

It looks to me like VFB4 is right on the edge of not having enough amplitude to switch logic states on the input pin of the controller.
Since the LEDs seem to be dying, I figured I would try to trick it into giving a bit more service. I tied FB1 and FB4 together at the LED connector on the main board. I figured this should make it operate as a 1 string LED panel and maybe the strong string would compensate for the weak one.
These are the waveforms now:
FB:

VFB:

This seems to have boosted the amplitude on all the VFB pins to the same level as VFB1 before the link was made.
The monitor seems to be working fine, but I will post an update tomorrow after being off overnight. Perhaps the temperature dependency I was seeing was just due to component tolerances changing with temperature and just pushing the already marginal signal over the threshold?
Let me know your thoughts.
EDIT: If anyone knows why my inline attachments aren't working, please let me know and I'll fix them.