EDIT: Now that I look closely, I have solved a different variant than what was asked. Probably this is interesting anyway :) The variant's rules are described here: http://www.chesscorner.com/tutorial/basic/pawngame/pawngame.htm
I have written a piece of software to solve this. Contrary to my and other people's expectations, and barring bugs, it shows that the game is actually a win for white. Without en passant it was a draw.
The only winning opening moves for white are 1. b4 and 1. c4 (and the symmetric 1. f4 and 1. g4).
Curiously, all other white opening moves are a win for black. 1. d4 is refuted by 1. ... b5 0-1; all other black responses are wins for white.
I put the code here: https://github.com/sliedes/pawnsonly. With the default setup it uses 25 gigabytes of memory for the transposition table and I seem to remember it takes a few hours to solve the 8x8 case. I have not tested how slow it is with less memory.
Here are some results of the first two or three moves:
Some of the lines end with "1/2-1/2+" or "1/2-1/2-". "1/2-1/2+" means that white can force at least a draw, and no further search was done because this information was enough for perfect play from the starting position (so it might also be a white win). Correspondingly, "1/2-1/2-" means that black is able to force at least a draw.