Fluke 863/867B Service Manual (4.5 MB)
The internal power supply of the 860 series multimeters is a DC-DC converter with multiple outputs and post regulators to provide the different voltages required by the various system components. The backlight is fed from the unregulated output of the DC-DC converter, controlled by a current sink circuit. I would expect that if there is a problem with the power supply of the backlight, this would also impact other power supply outputs. Since you do not mention any other issues with the meter, I'm tempted to assume the internal power supply is fine.
My guess is the problem is with the LED backlight on the Hitachi LMG7135PNFL LCD display. Bad contact or solder joint, maybe a failing LED, getting worse when heating up. I haven't inspected the LCD up closely, so I don't know whether the backlight is accessible/serviceable.
Hitachi stopped manufacturing these displays, but it seems there is still plenty of stock (at the moment of writing). For example:
https://www.aliexpress.com/item/32847956772.html (no affiliation - I just picked a random result from Aliexpress).
When in doubt, you can try to power the backlight without the multimeter connected. Sadly I could not find a datasheet for the LCD, but you can measure the voltage drop over the backlight (IBL+ and IBL- in the schematics) and according to the service manual the current through the backlight should be approximately 80 mA. If it flickers with an external power supply as well, you can be sure the problem is with the LCD module.