Check out Dr. Thomas's Ligonier lectures on Pilgrim's Progress. I'll have to check at home on which of the series covers this. As I recall, he favors the the idea that Pilgrim is saved at the Wicket Gate. I can think of two supports for this.
This is autobiographical, so the loss of the burden at the cross may represent Bunyan's being cured of an abiding sin.
In part 2, Christiana and company all enter through the Wicket Gate and the cross is just pointed out as where Christian lost his burden.