Well, sure, you don't lose the Bishop if you don't go all the way thru with the Noah's Ark Trap, but now you are just a flat pawn down, aren't you?  I've always heard that 5.d4 is a mistake against the Modern Steinitz, and that's what lead to the birth of lines like the Siesta Gambit (5.c3 f5).

Otherwise, based on your response, it sounds like maybe I am just over-estimating Black's compensation after exchanging on c6 and answering dxe5 with f6.  Being one that doesn't play the Schliemann, and have yet to face it over the board except in blitz doesn't help in my understanding of the Schliemann either, and so clearly there must be something about Black's position in the Schliemann that gives him the comp that I guess he just doesn't have here.

Thanks for the response.

So the main response to 3...d6 is 4.d4, yet after 3...a6 4.Ba4 d6, the Modern Steinitz, 5.d4 is a blunder, known as the Noah's Ark Trap!

In the books I've seen, they only seem to consider 4...Bd7 and 4...exd4 for Black.

What about 4...a6?  What should White do here?

A) Exchange on c6?
B) Retreat to c4?
C) Retreat to e2?
D) Counter-Attack the Knight with d5?

Playing 5.Ba4 just leads back to the Noah's Ark Trap, and 5.Bd3 just looks stupid, as does 5.Bf1 (White hasn't castled and gotten the Rook out yet to e1 in order to tuck the Bishop in on f1).

So what should White do here?  One might think that going with option A wins a pawn and trades Queens, leaving Black with doubled-isolated c-pawns, but does Black not get Schliemann-like Counterplay after 5.Bxc6 bxc6 6.dxe5 f6 instead of 6...dxe5, which is just bad?  Or am I overthinking this?

Would retreating to c4 be better?  Options c and d just look passive to me.

Any thoughts?