I don't recommend 2.e4. To me, the classical 2.Nf3 and 3.e3 plan seems best (Why needlessly sac a pawn when you can get a safe plus through normal methods anyway?). But then again it's a matter of taste so if you feel comfy in the final position of the main variation I've given then go for it!
Also note that you'll face 5...c5 only 1/10 times you play this, so it may be a good practical try after all.
I recently played this game as white
https://www.chess.com/game/live/42221458519
On 5th move, I played Nxd4 which is an inaccuracy and Bxc4 is the preferred move. Is it because this move develops a piece and allows casting on the next move. Or there is something deep? Thanks