Uhm, you'd never get the Traxler against me. You can't force it.
First off, White shouldn't always play 1.e4 and 2.Nf3. You mention the Caro-Kann. Against 1.e4 c6, White should play 2.d4, not 2.Nf3. Same thing goes for the French, Modern, Pirc, or any other opening that neither attacks e4 (1...d5/1...Nf6) nor d4 (1...c5, 1...e5).
Also, The Traxler assumes that White plays 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5. White can just as easily play 4.d4, or 3.d4/3.Bb5, or 2.f4/2.d4/2.Nc3, or even 1.d4/1.c4/1.Nf3.
So quit trying to find "sneaky" stuff, and learn the proper way to play chess. You may have to take your lumps as you learn an opening, but play something normal that meets your typical style of play and stick with it until you've mastered other parts of the game. Then expand.
I started out playing the French for 10 years - 1997 to 2007. Since then, I've played it sporadically, but I've also played 1...e5, Caro-Kann, Modern, Najdorf Sicilian, Taimanov Sicilian, and the Accelerated Dragon since then.
Not to mention, your post also makes no sense because you want to play the Traxler Counter-Gambit, but yet avoid the Scotch.
Uhm, after 1.e4 e5 2.Nf3 Nc6, what forces White to play 3.Bc4? Let's say I play 3.d4! Oops! Now we have a Scotch!
I can only say 1 thing about the original post:
SMH!
Sorry, this is probably old question...but is there a sneaky, attacking seldom used defense for Black against e4 then Nf3 by White?
Not the Petroff, a bit stale and boring.
Not the Scotch or Scotch Gambit, great but not sneaky enough.
Nothing like Italian or 4 N's. Not the Sicilian, too well known. Not the Philidor or Caro-Kann.
Something like the Traxler, that's what I've got in mind, but they don't often attack with the Fried Liver. (the Traxler is sneaky, surprising, bold, attacking seldom used or known below 1600; they think you've gone nuts and suicidal when you sac your Biship) (also could be similar to the Sokolsky except for Black, something totally insane looking but deceptively part sound) Is there anything that has all these? Doesn't have to be for Expert level, just for <1800