I just started using my new I2C LCD Interface v3 and, since I am using it with an old LCD module with no backlight, I have had no problems. There will be a serious problem if and when I attempt to use it with my newer backlit LCD module since you have made no provisions for a current limiting resistor. I realize that many of the newer LCD modules have a built in current limiting resistor but you should realize that many do not. I didn't notice this omission until I had already mounted the transistor. I suppose I can remove the transistor and put the current limiting resistor between it's C and E pads thus giving up software control of the backlight. Do you have a better suggestion since PWM is not an option when using an I2C extender.
It wouldn't be hard to incorporate such a resistor in your current board design but it would obliterate the silk-screen lettering for the push buttons unless you make the board longer. While you are at it you really should reconsider how you are implementing your backlight control. Specifically, since you are using high side switching you should be using a PNP transistor. This should be almost trivial to fix using your current board design.
Another improvement, which would probably require some extra real estate, would be some jumpers to enable/disable the I2C pull-up resistors. I have connected mine via some machined pins so I can remove them but jumpers would be nicer.