Why would there be a fight if Trump fired Cordray?

The Second Circuit Court of Appeals has already ruled that the section of the authorizing legislation that held that the Director of the CPFB could only be fired by Congress was unconstitutional.

The court held that the CPFB is a board under the aegis of the Executive Branch, and the President has the absolute right to fire or request the resignation of any Executive Branch employee that requires Senate confirmation.

Cordray is no different than Comey...He serves at the pleasure of the President.