I'm not sure how this would be a problem, I mean it could... The imaginary child would be the same as the real one.

For example, the passport photos would be the same picture of the same child. If it came down to getting caught doing this, the first thing a prosecutor would have to do is prove the flesh and blood version of both children existed in the first place. All they would have is the paper trail you created and that arguement could be countered simply by telling the truth. In order to discredit your identity farming story, they would need some sort of physical evidence I think. For example, there wouldn't be a record of the imaginary child ever enrolled in school, nor would there be any witnesses to claim they ever met the child.

I wonder which would be worse from a ciminal penalty standpoint, the supposed murder of a child, or years of social security fraud and falsifying various government documents? You could get tagged with multiple counts on the latter and rack up years and years in prison.

I suppose at the very least the murder charge, being a violent crime, would land you in maximum security whereas all the fraud charges would get you minimum security.