1.f4 (Bird opening) is the only one of the five “big” opening moves against which I do not recommend The Dark Knight System. This does not limit the use of 1… Nc6, because Bird is the least frequently played opening move, if not already “exotic” itself.
My recommendation is based on the following consideration, which follows from my own White game with 1.f4 . The Bird player does not necessarily want to play the modern system with fianchetto of the light square bishop (so-called Leningrad variant in the Bird system, analogous to the Leningrad system in the Dutch Defence), i.e. f4, g3, Bg2, Nf3 and 0-0. Rather, you fianchetto this bishop, because you do not really know what to do with him. The Bird player wishes to build up with f4, Nf3, e3, b3, Bb2, 0-0 – and at the latest before the short castling move, the Bf1 has to be moved somewhere, preferably to b5, where he exchanges himself for a black knight on c6, thus eliminating the problem piece and on top of that, the (key) square e5 has one less black defender.
But this setup only works if a knight on c6 appears in time – and TDKS is doing the White Player a favour with the first move.
Of course, you don’t lose with 1… Nc6 against 1.f4, but for the above mentioned principle reason I don’t recommend it. But if you hold the principle “1… Nc6 against really everything” higher, don’t let yourself be deterred by my recommendation.