The cases I cited made it abundantly clear that the courts were speaking of the President's "inherent constitutional authority", in others words, a power granted by the Constitution, and therefore s power that may not be infringed upon by Congress.
I'm only assuming that the court meant what it said when it held that the President did have inherent constitutional authority to conduct warrantless searches to obtain foreign intelligence information. And since, as the court held, this was a constitutional power of the President, it can't be infringed upon by Congress.
Actually, Truong did not speak of that. That was purely the invention of the court you quoted at the bottom. Truong (a case arising out of a pre-FISA action but decided after FISA was passed) was decided on the much narrower ground that ordinary courts lack the expertise to judge whether or not the probable-cause standard has been met in cases involving foreign surveillance. But it explicitly noted in Footnote 3 that FISA "will encourage the development of foreign intelligence expertise among these seven judges by empowering them to hear all foreign intelligence warrant requests." Thus satisfying the concerns they made in the case they were deciding.
This whole business of saying that the President has "inherent" powers that not even Congress can infringe on is completely specious. The Constitution gives the President not only explicit power but an explicit command to see to it that the laws are enforced, yet no one seriously disputes that Congress can regulate how this is done. And yet we're asked to believe, even "take it for granted", that powers that aren't explicitly granted to him by that document are somehow completely untouchable by Congress.