After a bit of thinking, I've solved it. It's quite a simple game. The player whose move makes the situation symmetrical will win. The player whose move breaks the symmetry will lose.
By "symmetrical" I mean a situation where both pairs of coins are the same distance away from each other. The starting setup is asymmetrical. The first player to move has a chance to make it symmetrical by moving their coin on the row of 10 squares so that there's 3 squares between the coins, just like on the short row. So the first player to move wins, assuming they know what they're doing. But if they don't make the perfect move, like leader here, then the advantage switches to the second player.