What seems to be strange from my point of view, that the display in Google Chrome - 500px (upload from the jpg out of Windows) is equal to the Lightroom version (tif).
This is really simple: if the data is color managed, it matches other color managed applications. If the data isn't color managed, there's no guarantee it will match and it usually doesn't.
Author “Color Management for Photographers" & "Photoshop CC Color Management/pluralsight"
If the monitor is wide gamut, just use Adobe RGB for a desktop background image. No special treatment required. It will look roughly right without color management (such as the Windows desktop) on a wide gamut monitor.
It will also look perfectly fine in color managed software everywhere and under all circumstances, as long as the profile is embedded. The Lightroom version in the OP is correct, and that's how it will look in all color managed applications everywhere.
Just don't use it for standard gamut monitors without full color management - then it will look dull and muted.
It may not be possible to cause your desktop background to become properly colour managed for showing photos. Others have now explained your options better than I can.
CCT Color Temp values define a pretty large number of possible colors and, everyone can calculate it differently. So I wouldn’t put much concern on specific values. This might help:
NOW, the issue in your enclosed PNG shows a color difference (not a WB difference per se) and I suspect that after you export the data, you are viewing it in a non color managed application. That is why it doesn't match LR which is color managed. Windows isn't color managing that data, so it's showing you an incorrect preview of the data!
Author “Color Management for Photographers" & "Photoshop CC Color Management/pluralsight"