increase the maximum brightness for a laptop replacement screen
you can only use the maximum brightness allowed by the manufacturer without taking it apart. Most of the laptop screens are LCD. For these, you might not be able to find a backlight brighter than normal. It is a meticulous task speaking from my experience.
I would rather suggest that you find an LED display. It has a lot more brightness and contrasts making it more readable outdoors. controllers and driver boards are available on eBay. First, make sure the driver board is available for that particular screen then choose the screen. Also, add a matt filter to avoid screen glare.