That requirement that Congress can't go on a fishing expedition only applies when they are NOT considering impeachment. During an impeachment proceeding, all immunity to investigation of the Executive Branch goes away. According to the Constitution, impeachment is an enumerated power of The House. In order to impeach, they have to have the tools of investigation. So this case is pretty moot now.

Also, the President has no power to say when an impeachment is legitimate. That is up to The House, and again according to the Constitution they get to make their own procedural rules.