I'll use my experience with the Southern and Freewill Baptists to speak to this, so bear that in mind when reading my replies.
When a Pastor comes to minister to the dying and their family, one of the first questions that they're taught to ask the dying is to confirm the state of their soul. No matter how long they've been a member of the Church, it's a question which is always asked. If there are doubts, the Pastor attempts to determine the root cause of the doubts and then addresses them, answering any and all questions. What I mean by root cause of the doubts is that the Pastor attempts to determine if the doubts are stemming from a wavering faith or if the doubts are originating from a lack of faith which has been, as you said, generated by a false conversion originating from a socially generated decision.
I know these things because I saw them ministered to my Mother when I was 23 and several of my grandparents over the years. It's very clearly understood, at least in the Churches which I have experience with, that this is the very last chance to make certain that everything is as it should be between the dying and God.
As are the last rites, and/or annointing of the sick.