I did read the story :)
In my opinion the relation between the Russian dill pickle and the whole story is this:
when the man was offered a picke by a russian coachman he said he felt it was natural and very right, you see. Then come the woman's thoughts: the pickle was sour. So I think it's just a symbol. He wants to share his feelings, his reminiscences with her, just feeling it's as right as sharing a dill pickle back then in Russia, but she does not want accept it because it is all painful for her or I don't know why...
This is just my wild guess of a dill pickle expert