Is it possible for DT024CTFT to perform "Write Display Brightness"?

Backlight is controlled by pins "A" and "K", pins 40 & 41, not through the ILI9341.

You have different ways to control the backlight:

1) No control over brightness, via a resistor hard wired.

2) A PWM signal generated by your micro, or through a light sensor.

Please refer to the CAT4139 for controlling backlight.