I think he's helping her because he finds a sense of purpose. After the first fight of the episode, you see Banagher working in the mech. The first shot makes him look as mechanical and lifeless as the mech itself.Originally Posted by einbreaker
He mentions it later that he "knows where he belongs" when he's working with the princess. Asking the princess to declare he's needed was his way of confirming and having someone acknowledge this purpose.
The last bit of the episode suggests that it's probably "in his blood" to stop wars/change futures etc like his deceased family before him.