How to control the display's backlight in the sketch?

since it will drop voltage even more so the LEDs won't be at full brightness?

It just doesn't work that way. The brightness of an LED depends on it's current. Similarly, the voltage across an LED also depends on it's current, but not linearly as with a resistor. You must start by considering the LED current, not the voltage.

I will do what dhenry said in his last post, and use PWM's max value instead of trying to limit it.

Think of it this way, this technique also protects things in case the PWM output goes to 100% because of an error in the programming.

Don