Logitech G13 changes LCD automatically, can it change the profile automatically, too?

The G13 doesn't actually know what environment you're in. The LCD content is not generated by the G13.

The game panel is simply outputting a bitmap provided to it by the game. What you are interpreting as the device switching modes is simply the game drawing a dramatically different picture to the G13's screen. The G13 remains unaware that a gameplay change has happened.