We already have a button ("½" on my keyboard) which swithes between different forms/sizes of the chat window.
How about ctrl-"½" which switches between two different chat contenst modes, one with and one without fight/dmg info. Then it will be possible, and very easy, after a fight to switch mode and see all the chat you missed during the fight. The game, of cause, have to have 2 history pages as it is not enough to remove the info from one to get the other.
I know there is an option to turn off fight/dmg info, but that is not an option for pet owners as it is the only way to see whether it fights or not (doing dmg).